Chapter 13.1, Problem 47E

To determine

To calculate: The number of ways to form a 9-player team.

Answer to Problem 47E

The number of ways to form a 9-player teamare $216216$ .

Explanation of Solution

Given information:

There are 3 players for catcher, 4 players who can play first base, 6 players who can only pitch and for remaining 6 positions 14 players are there. A 9-player team 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 3 players for catcher, 4 players who can play first base, 6 players who can only pitch and for remaining 6 positions 14 players are there. A 9-player team is to be formed.

In order to form a 9-player team, use combination as selection does not matter.

There are 3 ways to select a catcher, 4 ways to select a player who will play first base, 6 ways to select a player who can pitch.

Now, 3 positions out of 9 are filled to form a team.

Now, 14 players are available to fill 6 positions, so possible ways are $C\left(14,6\right)$ .

So, total number of required ways are $3\cdot 4\cdot 6\cdot C\left(14,6\right)$

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

Here n is 14 and r is 6for $C\left(14,6\right)$ .

  $\begin{array}{c}3\cdot 4\cdot 6\cdot C\left(14,6\right)=72\cdot \frac{14!}{6!\left(14-6\right)!}\\ =72\cdot \frac{14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8!}{6!\cdot 8!}\\ =72\cdot \frac{14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9}{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}\\ =216216\end{array}$

Thus, the number of ways to form a 9-player team are $216216$ .