Question

In a club with 12 male and 12 female​ members, how many 5-member committees can be chosen that have

​(a) at least 4 ​women?

​(b) no more than 2 men?

Answer 1

12 male and 12 female. And a 5-membered committee is to be formed.

a) Atleast 4 women means : either 4 or 5 women.

case1) So if we choose 4 women and 1 man then the number of ways are

  

case2) If we choose all 5 women then the number of ways are

So total number of ways is=5940+792=6732 answer

b) No more than two men means that either 2 or 1 or no men.

case3) If we take 2 men, then 3 will be women andthe number of ways will be

case4) If we take 1 man and 4 women, then it is similar to case1) and the number of ways is 5940.

case5) I we take no man, then we take all women which is similar to case2) and the number of ways is 792.

Hence total number of ways in this part is 14520+5940+792=21252 answer.