We need to choose a team of 11 from a pool of 15 players and also select a captain. Find the number of different ways this can be done.

We need to choose a team of 11 from a pool of 15 players and also select a captain. Find the number of different ways this can be done. 

(A) $\binom{15}{11}$, 

(B) $11 \times \binom{15}{11}$, 

(C) $15 \times 14 \times \cdots \times 5$, 

(D) $(15 \times 14 \times \cdots \times 5) \times 11$. 

Answer: (B) $11 \times \binom{15}{11}$ 

Explanation: 

Number of ways of selecting a captain from 15 players  = $\binom{15}{1}$. 

Remaining players in the pool $= 15 - 1  = 14$, and the remaining number of players to be chosen $= 11 - 1 = 10$. 

Number of ways of selecting 10 players from a pool of 14 players = $\binom{14}{10}$. 

Therefore, the total number of different ways of selecting 11 players from a pool of 15 players and also select a captain is 

$= \binom{15}{1} \times \binom{14}{10}$ 

$= 15 \times \frac{14 \times 13 \times 12 \times 11 \times 10!}{4! \times 10!}$ 

$= \frac{15 \times 14 \times 13 \times 12 \times 11!}{4! \times 11!} \times 11$

$= \binom{15}{11} \times 11$.