Question

In the $13$ cricket players $4$ are bowlers, then how many ways can form a cricket team of $11$ players in which at least $2$ bowlers are included?

• A. $55$
• B. $72$
• C. $78$
• D. None of these

Answer 1

The correct option is C

$78$

Explanation For The Correct Option:

Identifying the number of ways to form a team:

Total number of players $=13$

Total number of bowlers $=4$

Number of players we have to select $=11$

Number of players who are not bowlers = $=13-4=9$

Then the number of ways selecting a cricket team of $11$ players in which at least $2$ bowlers are included

Possibility $1$: selecting two bowlers and all $9$ players =${}^{4}C_2\times {}^{9}C_9$

Possibility $2$: selecting three bowlers and $8$ out of $9$ players= ${}^{4}C_3\times {}^{9}C_8$

Possibility $3$: selecting four bowlers and $7$ out of $9$ players = ${}^{4}C_4\times {}^{9}C_7$

$$\begin{gathered} \text{Number of ways}=\left({}^4{C}_2{\times }^9{C}_9\right)+\left({}^4{C}_3{\times }^9{C}_8\right)+\left({}^4{C}_4{\times }^9{C}_7\right) \\ =\left(\frac{4\times 3}{2}\right)\times \left(1\right)+\left(\frac{4\times 3\times 2}{3\times 2}\right)\times \left(\frac{9}{1}\right)+\left(1\right)\times \left(\frac{9\times 8}{2\times 1}\right)[\because {}^n{C}_r={}^n{C}_{n-r}\&{}^n{C}_0=1] \\ =6+36+36 \\ =78 \end{gathered}$$

Hence, the correct answer is option (C).