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

Keep in mind that we need to select at least four bowlers and the total should not exceed the total number of players.

Complete step-by-step answer:
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 can use the formula of combination, since the arrangement of members is not compulsory.

1. If we are choosing a team where it contains exactly 4 bowlers then it will contain 7 batsmen
   The number of possible ways = $\dfrac{6!}{4!(4-2)!}+ \dfrac{9!}{7!(9-7)!}$
2. If we choose a team where 5 bowlers exist then number of possible ways = $\dfrac{6!}{5!(6-5)!}+ \dfrac{9!}{6!(9-6)!}$
3. If we choose a team where 6 bowlers exist then the number of possible ways =$\dfrac{6!}{6!(6-6)!}+ \dfrac{9!}{5!(9-5)!}$
   Now, adding all the possibilities will give the total number of ways a team of 11 can be selected , so that it contains at least 4 bowlers .
   Total number of ways = $\dfrac{6!}{4!(4-2)!}+ \dfrac{9!}{7!(9-7)!}+ \dfrac{6!}{5!(6-5)!}+ \dfrac{9!}{6!(9-6)!}+ \dfrac{6!}{6!(6-6)!}+ \dfrac{9!}{5!(9-5)!}$
   = 15 + 36 + 6+ 84 + 1 + 126
   = 261
   Hence there are 261 ways.

Note: Students often go wrong in the sum by not reading the requirements carefully. We need to choose 11 players such that there is no repetition of either batsmen or bowlers. Students also need to keep in mind that the arrangement is not necessary as they are being chosen for a team.