2015/round1/winning_at_sports.html

<p>
In the game of <em>Sports</em>, the object is have more points than the other
team after a certain amount of time has elapsed. Scores are denoted by
two hyphen-separated integers. For example, scores may include 3-2,
4-1, or 10-0. The first number is how many points you've scored, and the second
is the number of points scored by the opposing team. You're very good at 
<em>Sports</em>, and consequently you always win. However, you don't always
achieve victory the same way every time.
</p>

<p>
The two most extreme kinds of victory are called <strong>stress-free</strong> and 
<strong>stressful</strong>. In a <strong>stress-free</strong> victory, you score the first
point and from then on you always have more points than your opponent.
In a <strong>stressful</strong> victory, you never have more points than your opponent
until after their score is equal to their final score.
</p>

<p>
Given the final score of a game of <em>Sports</em>, how many ways could you
arrange the order in which the points are scored such that you secure a
<strong>stress-free</strong> or <strong>stressful</strong> win?
</p>

<h3>Input</h3>

<p>
Input begins with an integer <strong>T</strong>, the number of games
you'll play. For each game, there is one line containing the final score of
the game in the format described above.
</p>


<h3>Output</h3>

<p>
For the <strong>i</strong>th game, print a line containing "Case #<strong>i</strong>: " 
followed by two space-separated integers, the number
of ways you can achieve a <strong>stress-free</strong> or <strong>stressful</strong> win,
respectively. Since these numbers may be very large, output them modulo
1,000,000,007.
</p>


<h3>Constraints</h3>
<p>
1 &le; <strong>T</strong> &le; 100 <br />
</p>

<p>
Since you always win, the first number in any final score
will always be larger than the second. Both scores will be non-negative 
integers not exceeding 2000.
</p>

<h3>Explanation of Sample</h3>
<p>
In the third test case, you can get a stress-free win by scoring points 1, 2, and 4, or points 1, 2, and 3.
You can get a stressful win by scoring points 2, 4, and 5, or points 3, 4, and 5.
</p>