id stringlengths 50 50 | setting_key stringclasses 1 value | setting_name stringclasses 0 values | messages listlengths 1 1 | ground_truth stringclasses 40 values | dataset stringclasses 1 value |
|---|---|---|---|---|---|
combinatory_probability_no_fixed_points_train_0000 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {y: 2, h: 1, j: 2, o: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 43 | math |
combinatory_probability_no_fixed_points_train_0001 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {q: 1, i: 2, m: 1, b: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0002 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {g: 1, a: 1, w: 1, f: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 37 | math |
combinatory_probability_no_fixed_points_train_0003 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {w: 1, k: 3, v: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 27 | math |
combinatory_probability_no_fixed_points_train_0004 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {m: 3, f: 1, q: 1, t: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
combinatory_probability_no_fixed_points_train_0005 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {q: 2, b: 2, j: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
combinatory_probability_no_fixed_points_train_0006 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {g: 1, i: 1, n: 2, r: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0007 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {z: 4, k: 1, x: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 113 | math |
combinatory_probability_no_fixed_points_train_0008 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {x: 2, q: 3, a: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0009 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {o: 1, n: 4, k: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 113 | math |
combinatory_probability_no_fixed_points_train_0010 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {s: 2, m: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
combinatory_probability_no_fixed_points_train_0011 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {g: 3, c: 2, x: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0012 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {n: 2, e: 1, u: 2, h: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 25 | math |
combinatory_probability_no_fixed_points_train_0013 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {w: 3, a: 4}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 9 | math |
combinatory_probability_no_fixed_points_train_0014 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {a: 3, x: 1, g: 1, e: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 59 | math |
combinatory_probability_no_fixed_points_train_0015 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {c: 3, l: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 13 | math |
combinatory_probability_no_fixed_points_train_0016 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {x: 1, h: 1, e: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 17 | math |
combinatory_probability_no_fixed_points_train_0017 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {b: 1, t: 1, n: 2, s: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 37 | math |
combinatory_probability_no_fixed_points_train_0018 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {b: 2, y: 1, m: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 17 | math |
combinatory_probability_no_fixed_points_train_0019 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {w: 1, f: 1, h: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
combinatory_probability_no_fixed_points_train_0020 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {p: 1, c: 2, r: 2, t: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
combinatory_probability_no_fixed_points_train_0021 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {y: 5, h: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 26 | math |
combinatory_probability_no_fixed_points_train_0022 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {f: 1, p: 1, e: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 27 | math |
combinatory_probability_no_fixed_points_train_0023 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {u: 1, h: 2, y: 1, c: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
combinatory_probability_no_fixed_points_train_0024 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {f: 2, s: 3, v: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 39 | math |
combinatory_probability_no_fixed_points_train_0025 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {u: 3, w: 2, j: 1, n: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 59 | math |
combinatory_probability_no_fixed_points_train_0026 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {l: 5, f: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 26 | math |
combinatory_probability_no_fixed_points_train_0027 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {t: 2, d: 1, f: 2, z: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
combinatory_probability_no_fixed_points_train_0028 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {n: 1, v: 2, g: 2, m: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 43 | math |
combinatory_probability_no_fixed_points_train_0029 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {a: 3, w: 1, g: 2, s: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 59 | math |
combinatory_probability_no_fixed_points_train_0030 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {t: 2, o: 1, l: 1, v: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0031 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {q: 2, g: 1, a: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 41 | math |
combinatory_probability_no_fixed_points_train_0032 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {d: 1, w: 2, h: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
combinatory_probability_no_fixed_points_train_0033 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {j: 2, g: 4}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 19 | math |
combinatory_probability_no_fixed_points_train_0034 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {v: 4, m: 1, e: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 16 | math |
combinatory_probability_no_fixed_points_train_0035 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {t: 2, h: 1, u: 2, i: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
combinatory_probability_no_fixed_points_train_0036 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {j: 2, o: 4}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 19 | math |
combinatory_probability_no_fixed_points_train_0037 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {x: 1, q: 1, f: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 11 | math |
combinatory_probability_no_fixed_points_train_0038 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {k: 2, z: 2, u: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 17 | math |
combinatory_probability_no_fixed_points_train_0039 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {s: 1, e: 1, a: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 17 | math |
combinatory_probability_no_fixed_points_train_0040 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {n: 1, m: 2, g: 1, v: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0041 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {c: 2, r: 2, q: 1, o: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
combinatory_probability_no_fixed_points_train_0042 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {r: 1, c: 1, u: 1, z: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 37 | math |
combinatory_probability_no_fixed_points_train_0043 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {n: 3, h: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 13 | math |
combinatory_probability_no_fixed_points_train_0044 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {q: 1, m: 1, y: 1, p: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 3 | math |
combinatory_probability_no_fixed_points_train_0045 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {g: 2, u: 2, e: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 39 | math |
combinatory_probability_no_fixed_points_train_0046 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {s: 1, t: 1, v: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
combinatory_probability_no_fixed_points_train_0047 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {s: 1, m: 1, o: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
combinatory_probability_no_fixed_points_train_0048 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {p: 3, z: 2, y: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0049 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {d: 1, e: 4}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
combinatory_probability_no_fixed_points_train_0050 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {r: 4, d: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 9 | math |
combinatory_probability_no_fixed_points_train_0051 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {j: 1, r: 2, i: 1, e: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0052 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {f: 1, x: 1, l: 2, r: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 37 | math |
combinatory_probability_no_fixed_points_train_0053 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {e: 2, z: 1, q: 1, g: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 37 | math |
combinatory_probability_no_fixed_points_train_0054 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {v: 1, c: 2, e: 1, r: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0055 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {u: 3, g: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 13 | math |
combinatory_probability_no_fixed_points_train_0056 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {x: 3, p: 1, k: 1, i: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 59 | math |
combinatory_probability_no_fixed_points_train_0057 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {i: 1, v: 1, s: 2, l: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 37 | math |
combinatory_probability_no_fixed_points_train_0058 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {b: 1, n: 4, k: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 16 | math |
combinatory_probability_no_fixed_points_train_0059 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {l: 1, e: 2, w: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 41 | math |
combinatory_probability_no_fixed_points_train_0060 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {y: 2, n: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
combinatory_probability_no_fixed_points_train_0061 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {i: 3, j: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 13 | math |
combinatory_probability_no_fixed_points_train_0062 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {o: 3, v: 1, u: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 11 | math |
combinatory_probability_no_fixed_points_train_0063 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {s: 1, n: 1, q: 1, i: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
combinatory_probability_no_fixed_points_train_0064 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {n: 3, g: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 13 | math |
combinatory_probability_no_fixed_points_train_0065 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {i: 1, r: 1, x: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 17 | math |
combinatory_probability_no_fixed_points_train_0066 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {j: 1, n: 3, r: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 41 | math |
combinatory_probability_no_fixed_points_train_0067 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {z: 2, q: 2, a: 1, f: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 43 | math |
combinatory_probability_no_fixed_points_train_0068 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {a: 1, f: 1, x: 1, o: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
combinatory_probability_no_fixed_points_train_0069 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {d: 1, g: 3, x: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 41 | math |
combinatory_probability_no_fixed_points_train_0070 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {l: 1, v: 1, x: 1, c: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0071 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {f: 3, u: 4}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 9 | math |
combinatory_probability_no_fixed_points_train_0072 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {l: 1, c: 1, i: 2, s: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 6 | math |
combinatory_probability_no_fixed_points_train_0073 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {n: 1, h: 2, q: 1, j: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 122 | math |
combinatory_probability_no_fixed_points_train_0074 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {e: 2, y: 3, p: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 11 | math |
combinatory_probability_no_fixed_points_train_0075 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {t: 1, d: 1, j: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
combinatory_probability_no_fixed_points_train_0076 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {h: 2, w: 5}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 26 | math |
combinatory_probability_no_fixed_points_train_0077 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {t: 1, s: 2, o: 2, a: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 43 | math |
combinatory_probability_no_fixed_points_train_0078 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {v: 2, g: 1, w: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 11 | math |
combinatory_probability_no_fixed_points_train_0079 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {p: 2, d: 2, h: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
combinatory_probability_no_fixed_points_train_0080 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {b: 1, w: 1, u: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 27 | math |
combinatory_probability_no_fixed_points_train_0081 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {l: 2, h: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 13 | math |
combinatory_probability_no_fixed_points_train_0082 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {i: 1, n: 1, j: 2, b: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 37 | math |
combinatory_probability_no_fixed_points_train_0083 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {b: 1, s: 1, i: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 27 | math |
combinatory_probability_no_fixed_points_train_0084 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {q: 1, g: 1, y: 2, b: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 37 | math |
combinatory_probability_no_fixed_points_train_0085 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {l: 1, r: 1, n: 2, c: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0086 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {k: 4, w: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 9 | math |
combinatory_probability_no_fixed_points_train_0087 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {k: 1, q: 1, a: 2, x: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 29 | math |
combinatory_probability_no_fixed_points_train_0088 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {h: 3, l: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 13 | math |
combinatory_probability_no_fixed_points_train_0089 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {g: 4, n: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 9 | math |
combinatory_probability_no_fixed_points_train_0090 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {o: 1, s: 1, p: 2, r: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 37 | math |
combinatory_probability_no_fixed_points_train_0091 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {g: 2, c: 1, e: 3, u: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 122 | math |
combinatory_probability_no_fixed_points_train_0092 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {g: 1, o: 2, b: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 17 | math |
combinatory_probability_no_fixed_points_train_0093 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {l: 3, j: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 13 | math |
combinatory_probability_no_fixed_points_train_0094 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {h: 2, t: 1, n: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
combinatory_probability_no_fixed_points_train_0095 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {r: 2, u: 4}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 19 | math |
combinatory_probability_no_fixed_points_train_0096 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {u: 3, t: 1, d: 1}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 11 | math |
combinatory_probability_no_fixed_points_train_0097 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {d: 2, p: 3, u: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 39 | math |
combinatory_probability_no_fixed_points_train_0098 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {z: 1, o: 5}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 7 | math |
combinatory_probability_no_fixed_points_train_0099 | combinatory_probability_no_fixed_points | null | [
{
"role": "user",
"content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {c: 2, q: 2}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}"
}
] | 4 | math |
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