Chapter 13.1, Problem 14CFU

To determine

To calculate: The number of different ways to form lineups of 9 players.

Answer to Problem 14CFU

The number of different ways to form lineups of 9 playersare $5005$ .

Explanation of Solution

Given information:

There are 15 members and lineups of 9 players is to be formed.

Formula used:

If there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Calculation:

Consider the provided information that there are 15 members and lineups of 9 players is to be formed.

In order to form lineups of 9 players, use combination as selection does not matter.

Recall that if there are n objects taken r at a time then combination is defined as $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ .

Required number of ways are $C\left(15,9\right)$

Here n is 15 and r is 9.

  $\begin{array}{c}C\left(15,9\right)=\frac{15!}{9!\left(15-9\right)!}\\ =\frac{15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9!}{9!\cdot 8!}\\ =\frac{15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10}{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}\\ =5005\end{array}$

Thus, the number of different ways to form lineups of 9 players are $5005$ .