Question

There are $6$ bowlers and $9$ batsmen in a cricket club. In how many ways can a team of $11$ be selected out of them, so that the team contains at least $4$ bowlers?

Answer 1

Given :

There are $6$ bowlers and $9$ batsmen.

The team of $11$ should be formed so that it contains at least 4 bowlers.

There are different ways to choose the team members.

We'll use combination formula, since the arrangement of members is not compulsory.

(i) If we are choosing a team where it contains exactly $4$ bowlers then it will contain 7 batsmen

$\therefore$ The number of possible ways $={}^6{C}_4\times {}^9{C}_7$

(ii) If we choose a team where $5$ bowlers exist then

Number of possible ways $={}^6{C}_5\times {}^9{C}_6$

(iii) If we choose a team where there are $6$ bowlers then

Number of possible ways $={}^6{C}_6\times {}^9{C}_5$

Now, adding all three possibilities will give the total number of ways a team of $11$ can be selected, so that it contains at least $4$ bowlers.

$\therefore$ total number of ways to select a team $={}^6{C}_4\times {}^9{C}_7+{}^6{C}_5\times {}^9{C}_6+{}^6{C}_6\times {}^9{C}_5$

$=\frac{6!}{2!4!}\times \frac{9!}{2!7!}+\frac{6!}{1!5!}\times \frac{9!}{3!6!}+\frac{6!}{6!}\times \frac{9!}{4!5!}$

$=15\times 36+6\times 84+1\times 126$

$=1170$

Hence there are total $1170$ ways.