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You can put this solution on YOUR website!
\n" ); document.write( "The answer is the number of solutions of the equation
\n" ); document.write( "
\n" ); document.write( "where a, b, and c are positive integers.
\n" ); document.write( "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.
\n" ); document.write( "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.
\n" ); document.write( "Now we use the stars and bars technique.
\n" ); document.write( "Represent the two remaining tokens with stars:
\n" ); document.write( "**
\n" ); document.write( "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.
\n" ); document.write( "For example,
\n" ); document.write( "*||* represents playing games 1 and 3 once each with the remaining 2 tokens;
\n" ); document.write( "|**| represents playing game 2 with both of the remaining tokens
\n" ); document.write( "The number of different ways of distributing the two remaining tokens is then the number of distinct ways of arranging the \"stars and bars\" ||**.
\n" ); document.write( "By a well known counting principle, that number of ways is
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\n" ); document.write( "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.
\n" ); document.write( "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:
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\n" ); document.write( "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.
\n" ); document.write( "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:
\n" ); document.write( "Find the number of terms in the simplified form of
\n" ); document.write( "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.
\n" ); document.write( "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
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