keneth rosen chapter 6 ex21

Question

A playoff between two teams consists of at most five games. The first team that wins three games
wins the playoff. In how many different ways can the playoff occur? (solution without manual calculation).

Answer 1

Ans will be $20$ ways

$1)$Here we can think, 1st three games win by winner , It can be done by either A or B $2$ ways

$2)$ Now Among first $4$ games

$4$ th one win by winner

Among $1$st  $3$ games $2$ wins by winner and $1$ win by loser=$\left ( \binom{3}{1}\times \binom{2}{2}=3 \right )$ ways

Now For $2$ player It can happen in $2$ ways i.e.$3\times 2=6$ ways

$3)$ First $4$ games makes a tie

and $5$ th one is winning game

So, choose first $4$ games as $\binom{4}{2}\times \binom{2}{2}=6$ ways

Now for both A and B it will be $6\times 2=12$ ways