Question

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

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

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* ^9P_9`

`\text{Possibilities}=(11!)/((11-9)!9!)* (9!)/((9-9)!)`

`\text{Possibilities}=(11* 10* 9!)/(2!9!)* (9!)/(0!)`

`\text{Possibilities}=(110)/(2* 1)* (9* 8* 7* 6* 5* 4* 3* 2* 1)/(1)`

`\text{Possibilities}=55* 362880`

`\text{Possibilities}=19958400`

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