There are five persons P, Q, R, S and T each one of whom has to be assigned one task. Neither P nor Q can be assigned Task-1. Task-2 must be assigned to either R or S. In how many ways can the assignment be done ?
- (a) 6
- (b) 12
- (c) 18
- (d) 24 ✓ UPSC's answer
Why the answer is (d)
• Five people take five tasks with one task each, so an assignment is a one-to-one matching of the five people to the five tasks.
• Task-2 must go to R or S, which splits the count into two cases.
• Take Task-2 = R. Task-1 can go to neither P nor Q, and R is already placed, so Task-1 is S or T — two choices. The remaining three people fill the remaining three tasks in 3! = 6 ways, giving 2 × 6 = 12.
• Task-2 = S is the mirror image of that case and also gives 12.
• The two cases are exclusive, so the total is 12 + 12 = 24, which is option (d).
Why the other options are wrong
- (a) 6
- 6 is just 3! — the arrangements of the three unconstrained people — and ignores that Task-1 and Task-2 can each be filled in more than one way.
- (b) 12
- 12 is the count for a single case, either Task-2 = R or Task-2 = S, and the question asks for both together.
- (c) 18
- 18 cannot arise from a correct split: the two cases are mirror images and therefore equal in size, so the total has to be even.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.