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A group of 10 people consists of 3 married couples and 4 si

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A group of 10 people consists of 3 married couples and 4 si

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A group of 10 people consists of 3 married couples and 4 single men. A committee of 4 is to be fo 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

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subh2273 wrote:
A group of 10 people consists of 3 married couples and 4 single men. A committee of 4 is to be fo 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
Good committees = total committees - bad committees.

Total committees:
From 10 people, the number of ways to choose 4 = 10C4 = (10*9*8*7)/(4*3*2*1) = 210.

Bad committees:
Since a good committee includes at most 1 married couple, a bad committee is composed of 2 married couples.
From 3 married couples, the number of ways to choose 2 to form a 4-member committee = 3C2 = (3*2)/(2*1) = 3.

Thus:
Good committees = 210-3 = 207.

The correct answer is B.

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Post
subh2273 wrote:
A group of 10 people consists of 3 married couples and 4 single men. A committee of 4 is to be fo 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 3C2 = 3 ways that 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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