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18 May 2019, 11:37
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A group of 10 people consists of 2 married couples and 6 bachelors. A committee of 4 is to be selected from the 10 people. How many different committees can be formed if the committee can consist of at most one married couple?
A) 95
B) 155
C) 185
D) 209
E) 210
A) 95
B) 155
C) 185
D) 209
E) 210
18 May 2019, 11:38
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dabaobao
A group of 10 people consists of 2 married couples and 6 bachelors. A committee of 4 is to be selected from the 10 people. How many different committees can be formed if the committee can consist of at most one married couple?
A) 95
B) 155
C) 185
D) 209
E) 210
A) 95
B) 155
C) 185
D) 209
E) 210
Official Solution
Credit: Veritas Prep
We have to select 4 people out of: 6 bachelors and 2 married couples.
The number of ways of selecting any 4 people out of 10 is 10*9*8*7/4! = 210 (Note here that we are just selecting 4 people. We are not arranging them so we divide by 4!)
The people will get selected in various ways:
1. Four bachelors
2. One from a couple and three bachelors
3. Two from two different couples and two bachelors
4. One couple and two bachelors
5. One couple, one person from a couple, one bachelor
6. Two couples
If we add the number of committees possible in each of these cases, we will get 210. Out of all these cases, only the last one (two couples) has more than one married couple. Instead of calculating the number of different committees that can be formed in each of the first five cases, we can calculate the number of committees in the last case and subtract it from 210.
How many different committees can be formed such that there are 2 couples? Only one since we have only 2 couples. We will have to select both the couples and we will get 4 people.
Number of different committees of 4 people such that there is at most one married couple = 210 – 1 = 209.
Just for practice, let’s see how we can calculate the different number of committees that can be formed in each of the first five cases. The sum of all these cases should give us 209.
1. Select 4 bachelors from 6 bachelors in 6*5*4*3/4! = 15 different committees
2. Select 1 person out of the two couples (4 people) in 4 ways and 3 bachelors from 6 bachelors in 6*5*4/3! = 20 ways. So you select the 4 people in 4*20 = 80 different committees
3. Select 2 people from 2 different couples in 4*2/2! = 4 ways and 2 bachelors from 6 bachelors in 6*5/2! = 15 ways. So you select the 4 people in 4*15 = 60 different committees
4. Select 1 couple in 2 ways and 2 bachelors from 6 bachelors in 6*5/2! = 15 ways. So you select the 4 people in 2*15 = 30 different committees
5. Select 1 couple in 2 ways, 1 person from the remaining couple in 2 ways and 1 bachelor from 6 bachelors in 6 ways. So you can select the 4 people in 2*2*6 = 24 different committees
The sum of all these five cases = 15 + 80 + 60 + 30 + 24 = 209 different committees (as expected)
ANSWER: D
Akshat1994
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Joined: 03 Aug 2019
Last visit: 19 Nov 2022
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Location: India
Schools: ISB '23 (A)
GMAT 1: 760 Q50 V41
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20 Sep 2020, 06:18
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The total ways in which a committee of 4/ 10 can be formed = 10C4
The no: of ways in which a committee can be formed with atmost one couple= Total ways- ( A committee of 4 with 2 couples)
Answer: 10C4-1= 209
Posted from my mobile device
The no: of ways in which a committee can be formed with atmost one couple= Total ways- ( A committee of 4 with 2 couples)
Answer: 10C4-1= 209
Posted from my mobile device
