Project ForgeThere are 4 women P, Q, R, and S, and 5 men V, W, X, Y, Z in a group. We are required to form pairs each consisting of one woman and one man. P is not to be paired with Z, and Y must necessarily be paired with someone. In how many ways can 4 such pairs be formed?
- A74
- B76
- 78
- D80
Explanation
4 of the 5 men get paired (one is left out); since Y must be paired, the leftover man is always one of V, W, X, or Z. Case leftover=Z: the remaining men V,W,X,Y (no Z, so P!=Z is automatically satisfied) pair with P,Q,R,S in 4!=24 ways. Case leftover=V (or W, or X): Z is among the paired men, so total pairings 4!=24 minus the sub-case P=Z (fixing that pair leaves 3!=6 ways for the rest) = 24-6=18 ways; this applies identically for leftover=V, W, or X. Total = 24 + 3x18 = 24+54 = 78.
Source: GATE ME 2017 Session 2 (ME-2) (2017)
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