GATE ME 2017 Set 2 — Question 63
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General Aptitude → Quantitative Aptitude → Permutations & Combinations
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Question
There are 4 women and 5 men in a group. We are required to form pairs each consisting of one woman and one man. is not to be paired with , and must necessarily be paired with someone. In how many ways can 4 such pairs be formed?
Correct answer
(C) 78
Solution
We need to form 4 pairs from 4 women and 5 men. All women must be paired.
Step 1: Total ways such that is paired.
can be paired with any of the 4 women: 4 ways.
Remaining 3 women can be paired with any 3 of the remaining 4 men ():
Number of ways = ways.
Total ways with paired = .Step 2: Subtract cases where is paired with (and is still paired).
If is a pair: 1 way.
must be paired with one of the remaining 3 women (): 3 ways.
Remaining 2 women must be paired with 2 of the remaining 3 men ():
Number of ways = ways.
Total restricted ways = .Step 3: Final calculation.
Total valid ways = .
Step 1: Total ways such that is paired.
can be paired with any of the 4 women: 4 ways.
Remaining 3 women can be paired with any 3 of the remaining 4 men ():
Number of ways = ways.
Total ways with paired = .Step 2: Subtract cases where is paired with (and is still paired).
If is a pair: 1 way.
must be paired with one of the remaining 3 women (): 3 ways.
Remaining 2 women must be paired with 2 of the remaining 3 men ():
Number of ways = ways.
Total restricted ways = .Step 3: Final calculation.
Total valid ways = .
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