Chapter 13.9, Problem 27PPS

To determine

To Find: No. of ways to win best of three, best of five and best of seven series.

Answer to Problem 27PPS

1. Number of ways to win a best of three series is $3$
2. Number of ways to win a best of five series is $10$
3. Number of ways to win a best of seven series is $35$

Explanation of Solution

Given information:

First team to win $2$ games wins the series.

Two ways to win bets of three series are to win the first $2$ games or lose the first game and win the next $2$ games.

Calculation:

There are two ways to win bets of three series are to win the first $2$ games or lose the first game and win the next $2$ games.

In calculating the number of ways of winning a best of three series, order is not important so arrangement sir combination.

No. of combinations is equal to no. of permutations divide by the no. of ways.

Calculate the no. of ways to win best of three series as below:

  $C\left(3,2\right)=\frac{P\left(3,2\right)}{No.ofwaysinwhicheachsetcanbearranged}$

There are $2.1=2$ ways of selecting the winning ways.

Now, calculate $P\left(3,2\right)$ ,

From $3$ games take $2$ at one time and calculate $3.2$

So,

  $\begin{array}{l}C\left(3,2\right)=\frac{3.2}{2.1}\\ =3\end{array}$

Number of ways to win a best of three series is $3$ .

For calculating the no. of ways of winning a best of five series, order is not important.

So arrangement is combination.

Number of combinations is equal to no. of permutations divided by no. of ways.

  $\begin{array}{l}C\left(5,2\right)=\frac{P\left(5,3\right)}{No.ofwaysinwhicheachsetcanbearranged.}\\ =3\end{array}$

There are $\mathrm{3.2.1}=6$ ways of choosing the winning ways.

Now, calculate $P\left(5,3\right)$ ,

From $5$ games take $3$ at one time and calculate $P\left(5,3\right)=\mathrm{5.4.3}$

So,

  $\begin{array}{l}C\left(5,3\right)=\frac{\mathrm{5.4.3}}{\mathrm{3.2.1}}\\ =10\end{array}$

Number of ways to win a best of five series is $10$ .

For calculating the no. of ways of winning a best of seven series, order is not important.

So arrangement is combination.

Number of combinations is equal to no. of permutations divided by no. of ways.

  $\begin{array}{l}C\left(7,4\right)=\frac{P\left(7,4\right)}{No.ofwaysinwhicheachsetcanbearranged.}\\ =24\end{array}$

There are $\mathrm{4.3.2.1}=24$ ways of choosing the winning ways.

Now, calculate $P\left(7,4\right)$ ,

From $7$ games take $4$ at one time and calculate $P\left(7,4\right)=\mathrm{7.6.5.4}$

So,

  $\begin{array}{l}C\left(7,4\right)=\frac{\mathrm{7.6.5.4}}{\mathrm{4.3.2.1}}\\ =35\end{array}$

Number of ways to win a best of seven series is $35.$