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Sagot :
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. [tex]^{11}C_9[/tex].
Second, we need to arrange the order of 9 selected players, it involves permutation, i.e., [tex]^9P_9[/tex].
The number of possibilities for the given scenario is:
[tex]\text{Possibilities}=^{11}C_9\times ^9P_9[/tex]
[tex]\text{Possibilities}=\dfrac{11!}{(11-9)!9!}\times \dfrac{9!}{(9-9)!}[/tex]
[tex]\text{Possibilities}=\dfrac{11\times 10\times 9!}{2!9!}\times \dfrac{9!}{0!}[/tex]
[tex]\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}[/tex]
[tex]\text{Possibilities}=55\times 362880[/tex]
[tex]\text{Possibilities}=19958400[/tex]
Therefore, the following scenario involves both permutation and combination, and the number of possibilities is 19958400.
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