A committee of 5 people is to be formed from a group of 8 men and 4 wo

Question

A committee of 5 people is to be formed from a group of 8 men and 4 women. How many different committees can be formed if the committee must include at least 2 women?

A. 500
B. 504
C. 540
D. 456
E. 600

Moyen · Answer

To solve this problem, we use combinations and consider the cases where the committee includes at least 2 women. The total group consists of 8 men and 4 women.

The committee must have at least 2 women, so we consider the following cases:

Case 1: 2 women and 3 men
 4C2 × 8C3 = 336

Case 2: 3 women and 2 men

4C3 × 8C2 = 112

Case 3: 4 women and 1 man

4C4 × 8C1 = 8

Step 2: Add all the cases

336 + 112 + 8 = 456

Final Answer: D. 456

blonpina · Answer

No constraints - 12C5 = 792
No women - 8C5 = 56
Only 1 women - 8C4 * 4 = 280
At least 2 women = 792 - 56 - 280 = 456

arthurbianchi · Answer

Hi Blonpina, I was following the same logic as you but got lost in the 8C4*4, did you multiply it by 4 because one of the slots out of the 5 was already taken by one of the 4 women?

Thanks for the help

blonpina · Answer

Hi arthurbianchi!

There are 2 parts:
1) 8C4 - the number of ways 8 men can take 4 slots (no women involved so far, we left 1 vacant slot for one)
2) * 4 - the number of ways 1 woman can fill the slot that we left in above equation (remember we filled 4 out of 5). As there are 4 women, you multiply it by 4

bruhletmein · Answer

Approach I took:

Ways of selecting at least 2 W = Total - Ways of selecting 1 W - Ways of selecting no women

Total = 12C5 = 12!/7!5! = 792

Ways of selecting 1 W (and 4 M) = 4C1 * 8C4 = 280

Ways of selecting no W (and 5M) = 8C5 = 56

Answer = 792 - 280 - 56 -> Not the best idea I did but basically saw 2 at the end of 792 and 6 at end of 56, meaning answer will end in 6

D is the only one that fits

A committee of 5 people is to be formed from a group of 8 men and 4 wo

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A committee of 5 people is to be formed from a group of 8 men and 4 wo

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