Question 1145099
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The answer is the number of solutions of the equation<br>
{{{a+b+c=25}}}<br>
where a, b, and c are positive integers.<br>
Problems like that can be solved using a method popularly known as "stars and bars".  Let's look at solving your problem using this method where the number of tokens is 5 instead of 25.<br>
We start by assigning 1 token to each of the 3 games.  That done, we have satisfied the requirement of playing each game at least once, and we are now left with 2 tokens to use on any games we want.<br>
Now we use the stars and bars technique.<br>
Represent the two remaining tokens with stars:<br>
**<br>
To represent dividing those two tokens among the 3 games, use two separator symbols "|"; the two separator symbols will divide the stars into 3 groups.<br>
For example,
*||*  represents playing games 1 and 3 once each with the remaining 2 tokens;
|**|  represents playing game 2 with both of the remaining tokens<br>
The number of different ways of distributing the two remaining tokens is then the number of distinct ways of arranging the "stars and bars" ||**.<br>
By a well known counting principle, that number of ways is<br>
{{{4!/((2!)(2!)) = C(4,2) = 6}}}<br>
So in our smaller problem, the number of ways of using 5 tokens to play the three games, playing each game at least once, is 6.<br>
For your problem, with 25 tokens, we again use 3 of them to make sure we play each of the 3 games at least once.  Then we are left with 22 tokens (stars) to be divided among 3 games using 2 divider symbols (bars); and the number of ways to divide the 25 tokens among the 3 games playing each game at least once is determined using 25-3=22 stars and 2 bars:<br>
{{{24!/(22!)(2!) = 24C2 = 276}}}<br>
The stars and bars method can be used on a large number of problems where the solution can be modeled by an equation of the form a+b+c+...=N where N is an integer total and the variables a, b, c,... are non-negative integers.<br>
The most common other kind of problem like this, in my experience, is finding the number of terms in the expansion of a polynomial.  As an example, of that, here is a problem that uses the same numbers as in your problem:<br>
Find the number of terms in the simplified form of {{{(x+y+z)^22}}}<br>
In each term of the expansion, the exponents are all integers, and the sum of the exponents is 22.  So again we have a problem where the answer is the number of non-negative integer solutions of the equation a+b+c=22.<br>
So again we have a case of 22 "stars" (the exponents) being divided into 3 groups (the variables x, y, and z) by 2 separator symbols ("bars").  And so, like your problem, the answer to this problem would be<br>
{{{24C2 = 276}}}