{
  "problem_id": "math_3",
  "domain": "math",
  "difficulty": "medium",
  "prompt": "A fair six-sided die is rolled 4 times. What is the probability that the sum of the rolls is exactly 15? Show all your work.",
  "num_total_tokens": 4096,
  "num_thinking_tokens_approx": 4095,
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  "thinking_text": "\nOkay, so I need to find the probability that the sum of four rolls of a fair six-sided die is exactly 15. Let me think about how to approach this. \n\nFirst, probability problems often involve figuring out the number of favorable outcomes divided by the total number of possible outcomes. In this case, the total number of possible outcomes when rolling a die four times is straightforward. Since each die has 6 sides, and each roll is independent, the total number of outcomes is 6^4. Let me calculate that: 6*6*6*6. 6*6 is 36, 36*6 is 216, 216*6 is 1296. So there are 1296 possible outcomes. \n\nNow, the tricky part is figuring out how many of these outcomes result in a sum of 15. This seems like a problem that can be approached using combinatorics, specifically the stars and bars method or generating functions. But I need to remember that each die can only show a number between 1 and 6. So I have to account for the constraints on each die's value. \n\nLet me recall the stars and bars theorem. The general idea is that the number of ways to distribute n identical items into k distinct bins with each bin having at least one item is C(n-1, k-1). If there are no restrictions (i.e., bins can be zero), it's C(n + k -1, k -1). But here, each die must show at least 1 and at most 6. So maybe I can use stars and bars with inclusion-exclusion?\n\nAlternatively, generating functions might be more straightforward. The generating function for a single die is x + x^2 + x^3 + x^4 + x^5 + x^6. For four dice, the generating function would be (x + x^2 + x^3 + x^4 + x^5 + x^6)^4. The coefficient of x^15 in this expansion would give the number of ways to get a sum of 15. \n\nHmm, so maybe I can compute that coefficient. But expanding that polynomial manually would be tedious. Let me think if there's a smarter way. \n\nAlternatively, I can model this as an integer solution problem. Let me denote each die roll as a variable: let x1, x2, x3, x4 be the results of each die. Each xi is an integer between 1 a",
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