Bunuel wrote:
A group of 10 people consists of 3 married couples and 4 single men. A committee of 4 is to be formed from the 10 people. How many different committees can be formed if the committee can consist of at most 1 married couple?
(A) 105
(B) 207
(C) 210
(D) 540
(E) 5,040
If there are no restrictions on how to select the 4 people from a group of 10 people, then we have:
10C4 = 10!/[4!(10-4)!] = 10!/(4!6!) = (10 x 9 x 8 x 7)/4! = (10 x 9 x 8 x 7)/(4 x 3 x 2 x 1) = 5 x 3 x 2 x 7 = 210
ways to select them.
All of these ways will consist of at most 1 married couple, except if the 4 people picked consist of 2 married couples. So, let’s determine the number of ways 2 married couples can be picked as a committee of 4:
If a committee of 4 consists of 2 married couples, then it could be: (couple 1, couple 2), (couple 1, couple 3), or (couple 2, couple 3). Thus, there are only 3 ways 2 married couples can be picked for the committee of 4. Subtract this from 210 and we have 207 ways to select a committee of 4 that will consist of at most 1 married couple.
Answer: B
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