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A group of 10 people consists of 3 married couples and 4 single men. A

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A group of 10 people consists of 3 married couples and 4 single men. A  [#permalink]

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New post 06 Jun 2016, 04:53
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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 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

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Re: A group of 10 people consists of 3 married couples and 4 single men. A  [#permalink]

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New post 06 Jun 2016, 07:32
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Total number of possibilities of selecting 4 people out of 10 without any restrictions = 10C4 = 10!/(4!*6!)
= (10*9*8*7)/4*3*2*1
= 10*3*7
= 210
Number of possibilities of selecting 2 married couples = 3C2 = 3!/2! = 3
Number of different committees with atmost 1 married couple that can be formed = 210 - 3 = 207
Answer B
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Re: A group of 10 people consists of 3 married couples and 4 single men. A  [#permalink]

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New post 05 Sep 2016, 23:59
Skywalker18 wrote:
Total number of possibilities of selecting 4 people out of 10 without any restrictions = 10C4 = 10!/(4!*6!)
= (10*9*8*7)/4*3*2*1
= 10*3*7
= 210
Number of possibilities of selecting 2 married couples = 3C2 = 3!/2! = 3
Number of different committees with atmost 1 married couple that can be formed = 210 - 3 = 207
Answer B


Shouldn't the possibilities of selecting 2 married couples be 4C2 here? Please explain.
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Re: A group of 10 people consists of 3 married couples and 4 single men. A  [#permalink]

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New post 06 Sep 2016, 00:13
ranaazad wrote:
Skywalker18 wrote:
Total number of possibilities of selecting 4 people out of 10 without any restrictions = 10C4 = 10!/(4!*6!)
= (10*9*8*7)/4*3*2*1
= 10*3*7
= 210
Number of possibilities of selecting 2 married couples = 3C2 = 3!/2! = 3
Number of different committees with atmost 1 married couple that can be formed = 210 - 3 = 207
Answer B


Shouldn't the possibilities of selecting 2 married couples be 4C2 here? Please explain.


There are only 3 married couples. Number of ways 2 couples can be selected out of 3 is 3C2.
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Re: A group of 10 people consists of 3 married couples and 4 single men. A  [#permalink]

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New post 04 Oct 2017, 16:40
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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Re: A group of 10 people consists of 3 married couples and 4 single men. A  [#permalink]

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New post 01 Nov 2017, 01:15
Skywalker18 wrote:
Total number of possibilities of selecting 4 people out of 10 without any restrictions = 10C4 = 10!/(4!*6!)
= (10*9*8*7)/4*3*2*1
= 10*3*7
= 210
Number of possibilities of selecting 2 married couples = 3C2 = 3!/2! = 3
Number of different committees with atmost 1 married couple that can be formed = 210 - 3 = 207
Answer B


Why are we subtracting the number of married couple from total? Why not add them to 210? Would be happy to know the reasoning behind this:)
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A group of 10 people consists of 3 married couples and 4 single men. A  [#permalink]

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New post 21 Feb 2018, 20:24
Madhavi1990 wrote:
Skywalker18 wrote:
Total number of possibilities of selecting 4 people out of 10 without any restrictions = 10C4 = 10!/(4!*6!)
= (10*9*8*7)/4*3*2*1
= 10*3*7
= 210
Number of possibilities of selecting 2 married couples = 3C2 = 3!/2! = 3
Number of different committees with atmost 1 married couple that can be formed = 210 - 3 = 207
Answer B


Why are we subtracting the number of married couple from total? Why not add them to 210? Would be happy to know the reasoning behind this:)



We are subtracting the number of married couple from the total because we want to form a committee that has at most 1 married couple.
The total number of ways by which we can form a committee of 4 from 10 people(4 single and 3 couples) is 210. These 210 ways can have different combinations of single men and couples.

But the number of couples should not exceed 1, that is it should not be 2. So in how many ways it can exceed 1 = 3C2. Meaning from 3 couples we are selecting 2 couples for the committee such that the committee will have all couples and no single men. We want this case excluded. Therefore 210 - 3C2 = 210 - 3 = 207.
A group of 10 people consists of 3 married couples and 4 single men. A &nbs [#permalink] 21 Feb 2018, 20:24
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