Can someone help me solve this GMAT Quant question?

Question

A committee of 4 members is to be formed from a group of 7 men and 5 women. How many different committees can be formed if the committee must include at least 2 women?
I tried listing out the possible combinations, but I'm getting confused with the restrictions. How should I approach this problem?

V Ritik · Accepted Answer

The key is to break it into cases based on how many women are in the committee, since it says "at least 2 women."
You have 5 women and 7 men total. The committee has 4 members. The possible cases are:
Case 1: 2 women, 2 men Ways to pick 2 women: C(5,2) = 10 Ways to pick 2 men: C(7,2) = 21 Total for this case: 10 &times; 21 = 210
Case 2: 3 women, 1 man Ways to pick 3 women: C(5,3) = 10 Ways to pick 1 man: C(7,1) = 7 Total: 10 &times; 7 = 70
Case 3: 4 women, 0 men Ways to pick 4 women: C(5,4) = 5 Total: 5
Now add them up: 210 + 70 + 5 = 285
So, 285 different committees can be formed.

Ranjan J · Answer

I struggled with these too, but once you break it into smaller parts, it gets easier. Since the committee needs at least 2 women, just handle it case by case:

For 2 women: choose 2 from 5 &rarr; C(5,2) = 10; choose 2 men from 7 &rarr; C(7,2) = 21; total = 10 &times; 21 = 210.
For 3 women: choose 3 from 5 &rarr; C(5,3) = 10; choose 1 man from 7 &rarr; C(7,1) = 7; total = 10 &times; 7 = 70.
For 4 women: choose 4 from 5 &rarr; C(5,4) = 5; no men needed; total = 5.
Add them: 210 + 70 + 5 = 285. The answer is 285 possible committees.

Once you set up the cases properly, these questions mostly become simple combinations.