In a room there are 5 couples, in how many ways two men and two women

Question

In a room there are 5 couples, in how many ways two men and two women can be selected such that exactly one couple is selected?

A. 30
B. 50
C. 60
D. 90
E. 120

Feb2024 · Answer

First, let us choose one couple out of the five to have exactly 1 man and 1 woman chosen.
This can be done in  5C1  ways, which is equal to 5.

Next, choose 1 man from the remaining 4 couples in  4C1  ways.

Next, choose exactly 1 woman from the remaining 3 couples in  3C1  ways.

Thus, the total number of ways of choosing two men and two women such that exactly one couple is selected =  5C1*4C1*3C1 = 60 ways .

So, C is the correct choice.

Note that we could also have first chosen a woman (5C1), then a man (4C1) and finally a couple (3C1) and our answer would still be the same. This is because here the order of choosing an individual or a pair is immaterial.

Ayeka · Answer

We can choose any one couple out of 5, so 5C1 
From remaining 4 couples, we can choose a man in 4C1 ways.
A woman can not be chosen in 4C1 as she might be the wife of the man selected and we will get a second couple. So, we can choose a women in 3C1 ways.

Total number of ways of choosing 2 man & 2 women such that exactly one couple is selected= 5C1*4C1*3C1
= 5*4*3
=60 ways

ANSWER: C

Posted from my mobile device

stevegv · Answer

I approached the question a little differently from the earlier replies.

i) First, there are 5C1 ways of selecting exactly one couple from the 5 couples - we have now successfully selected exactly one couple.
ii) Then, to ensure that the next man and woman are from different couple pairs, we pick two different couples from the remaining 4 couples (4C2), out of which we will further select the next man and woman in steps iii) and iv).
iii) Now, there are 2C1 ways of selecting a man from the first couple obtained from the 4C2 selection in step ii).
iv) After that, there remains only 1 way to select a woman from the second couple obtained from our 4C2 selection in step ii).

Thus giving us -> 5C1 * 4C2 * 2C1 * 1 = 5 * 6 * 2 = 60

Answer: C. 60

Vsolo · Answer

5C1 = 5 ; no of ways of selecting exactly one couple.

Remaining 2 places should be filled by a man and woman. Let's assume the question imposes no conditions. Then we have 4 (Men) x 4 (Women) ways of filling the remaining 2 places. Therefore 16 ways.

Let's now remove the exclusions from 16. There are 4 exclusions as each of the 4 men go with 4 distinct women and vise-versa. Therefore 16-4 = 12.

Now we have 5C1 x 12 ways as the case which satisfies the condition given in the question. Therefore answer is 5 x 12 = 60.  Option C

In a room there are 5 couples, in how many ways two men and two women

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In a room there are 5 couples, in how many ways two men and two women

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