Question

Decide if the following scenario involves a permutation or combination. Then find the number of possibilities.

Answer 1

Given:

The batting order for nine players on a team with 11 people.

To find:

The number of possibilities for the given scenario.

Solution:

First, we need to select 9 players from 11 people, it involves combination, i.e. $^{11}C_9$.

Second, we need to arrange the order of 9 selected players, it involves permutation, i.e., $^9P_9$.

The number of possibilities for the given scenario is:

$\text{Possibilities}=^{11}C_9\times ^9P_9$

$\text{Possibilities}=\dfrac{11!}{(11-9)!9!}\times \dfrac{9!}{(9-9)!}$

$\text{Possibilities}=\dfrac{11\times 10\times 9!}{2!9!}\times \dfrac{9!}{0!}$

$\text{Possibilities}=\dfrac{110}{2\times 1}\times \dfrac{9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{1}$

$\text{Possibilities}=55\times 362880$

$\text{Possibilities}=19958400$

Therefore, the following scenario involves both permutation and combination, and the number of possibilities is 19958400.