Question

A committee of 5 people is to be formed from 6 men and 4 women. Find the number of these selections with more women than men.​

Answer 1

Step-by-step explanation:

The number of ways is 186 , as noted in other answers. Here’s another way to do it: take all possible ways of choosing five people, and subtract the ways that are all five men or four men and one woman: (105)−(65)(40)−(64)(41)=186.

But more importantly, an answer to your second question. The reason (42)(83) is wrong is because you double count certain combinations. Remember when you multiply (42)=6 and (83)=56 and claim that gives you the answer, you’re saying for each of the 6 ways of choosing two women, there are 56 ways of choosing the rest, each of which results in a unique combination.

This is clearly false. Say you choose Alice and Becky, but when you choose the remaining three, Daisy finds her way in (along with Jack and John). This combination is double counted, for instance, when you choose Alice and Daisy, but Becky (and Jack and John) are selected from the remaining