Question

There are 12 volleyball players 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

Hint- In this question it is given that the captain always remains the same after selecting players. So we will form a new team with remaining players by using the simple concept of combination. Use the basic formula for selecting n number of items from a set of m items for solving the problem.

Complete step-by-step answer:
Given that: in college there are 12 volleyball players including captain, we have to select 9 players out of 12 but it is given that captain remains the same.
As the position of captain is fixed
Now we have to select only 8 volleyball players out of 11 players.
We know that when we have to select “m” items from given “n” total numbers of things, it can be selected in ${}^n{C_m}$ ways.
So, total number of ways of selection of 8 players out of 11 players is:
$ = {}^{11}{C_8}$
Also we know the formulas for combination:
${}^x{C_y} = {}^x{C_{x - y}}$
So simplifying above term, using the given formula we get:
No of ways of selection is:
$
   = {}^{11}{C_8} \\
   = {}^{11}{C_{11 - 8}} \\
   = {}^{11}{C_3} \\
 $
Let us further simplify the term
\[
   \Rightarrow {}^{11}{C_3} = \dfrac{{11!}}{{3!\left( {11 - 3} \right)!}}{\text{ }}\left[ {\because {}^x{C_y} = \dfrac{{x!}}{{y!\left( {x - y} \right)!}}} \right] \\
   = \dfrac{{11!}}{{3!8!}} \\
   = \dfrac{{9 \times 10 \times 11}}{{3 \times 2 \times 1}} \\
   = \dfrac{{990}}{6} \\
   = 165 \\
 \]
Therefore, there are 165 ways in which the team can be formed.
Hence, the correct answer is option D.

Note- The method of permutation and combination is used for solving such practical problems. Permutation and combination are the ways to represent a group of objects by selecting them in a set and forming subsets. It defines the various ways to arrange a certain group of data. When we select the data or objects from a certain group it is said to be permutations whereas the order in which they are represented is called combination. Both concepts are very important in Mathematics.