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## Combinatorics

Yesterday in Pre-Calculus class we were discussing combinatorics and came across a series of questions on how many ways can a best 2 out of 3 sets tennis match be won and how many ways can a best 3 out of 5 tennis match be won.

For a 2 out of 3 scenario there are six possibilities (AA, ABA, BAA, BB, BAB, ABB) and for the 3 out of 5 scenario there are 20 possibilities (AAA, BAAA, ABAA, AABA, BBAAA, BABAA, BAABA, ABBAA, ABABA, AABBA and ten more in which B wins).

The question then came up of how to generalize these results.  The way I approached this was to consider how many ways one of the players could win and then double the answer to include the scenarios in which the other won.

One of the issues in this type of problem is that once the winner has won the required number of games the match ends, so that a best 2 out of 3 match could never end AAB, for instance.  Another issue is that each of the possibilities of winning in a different number of sets should be considered separately.  That is, winning in the minimum number of sets, then winning after having lost one set and so forth.

Taking all of these issues into account for a competition in which the winner must win $\frac{n+1}{2}$ out of $n$ leads to a general formula of $2*\displaystyle \sum_{k=0}^{\frac{n-1}{2}}{_\frac{n+2k-1}{2}}C_{k}$.

Or – if the winner must win $g$ out of $2g-1$ then the formula would be $2*\displaystyle \sum_{k=0}^{g-1}{_{g+k-1}}C_{k}$.