Question

The number of ways in which a cricket team of 11 players be chosen out of a batch of 15 players so that the captain of the team is always included is,
(a) 165
(b) 364
(c) 1001
(d) 1365

Answer 1

Hint: We have to choose 11 players from the 15 players such that the captain is always included. As captain is always included so we don’t have to choose 1 player i.e. captain out of 15 players. Now, 14 players are remaining to choose from and captain is already selected so we have to select 10 players from 14 players which can be selected through combinatorial method i.e. ${}^{14}{{C}_{10}}$.

Complete step-by-step answer:
It is given that the 11 players cricket team contains a captain. So, we don’t have to select a captain from 15 players because it is always included. From 15 players, one player is excluded i.e. captain so we are left with 14 players. Now, from 11 players’ cricket team, captain is always included so we have to select 10 players from the batch of 14 players.
Using a combinatorial method we can choose 10 players out of 14 players.
${}^{14}{{C}_{10}}$
To expand the above combinatorial symbol we are using the following formula:
${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$
To find the value of ${}^{14}{{C}_{10}}$ we are going to substitute n as 14 and r as 10 in the above equation.
${}^{14}{{C}_{10}}=\dfrac{14!}{10!\left( 14-10 \right)!}$
$\Rightarrow {}^{14}{{C}_{10}}=\dfrac{14!}{10!\left( 4 \right)!}$
We can write the 14! Written in the numerator of the above equation as 14.13.12.11.10!.
${}^{14}{{C}_{10}}=\dfrac{14.13.12.11.10!}{10!\left( 4 \right)!}$
In the above equation, 10! will be cancelled out from the numerator and denominator and we are left with:
$\begin{aligned} & {}^{14}{{C}_{10}}=\dfrac{14.13.12.11}{4.3.2.1} \\\ & \Rightarrow {}^{14}{{C}_{10}}=\dfrac{14.13.12.11}{24} \\\ & \Rightarrow {}^{14}{{C}_{10}}=1001 \\\ \end{aligned}$
From the above calculation, the number of ways of choosing 11 players from the batch of 15 players such that the captain is always included is 1001.
Hence, the correct option is (c).

Note: The possible mistake that could happen is you will choose 11 players out of 15 players in the following wrong ways:
By including the captain only 10 players we have to select out of 15 which is equal to:
${}^{15}{{C}_{10}}$
As captain is included in the 11 players so 1 player from 15 players is already selected and you might have chosen 11 players out of 14 players in the following way:
${}^{14}{{C}_{11}}$
So, beware of choosing 11 players from the batch of 15 players.