Question

There are $12$ volleyball players in all in a college, out of which a team of $9$ players is to be formed. If the captain always remains the same, then in how many ways can the team be formed

• A. $36$
• B. $108$
• C. $99$
• D. $165$

Answer 1

The correct option is D

$165$

Compute the total number of ways:

There are $12$ volleyball players and we need to create a team of $9$ players.

But the captain always remains the same, so we need to select $8$ players out of $11$.

Therefore, the total number of ways is

$$\begin{gathered} {}^{11}\mathrm{C}_8\left[{}^{\mathrm{n}}\mathrm{C}_{\mathrm{r}}=\frac{\mathrm{n}!}{\mathrm{r}!\times (\mathrm{n}-\mathrm{r})!}\right] \\ =\frac{11!}{8!\times 3!} \\ =\frac{11\times 10\times 9\times 8!}{8!\times 3\times 2} \\ =165 \end{gathered}$$

Hence, option (D) is the correct option.