Three persons A, B and C are standing in a queue not necessarily in the same order. There are 4 persons between A and B, and 7 persons between B and C. If there are 11 persons ahead of C and 13 behind A, what could be the minimum number of persons in the queue?
- (a) 22 ✓ UPSC's answer
- (b) 28
- (c) 32
- (d) 38
Why the answer is (a)
• Let the positions of A, B, and C be $P_A$, $P_B$, and $P_C$ respectively, with position 1 being the front of the queue.
• The condition '11 persons ahead of C' implies $P_C = 12$.
• The condition '13 persons behind A' implies $P_A = N - 13$, where $N$ is the total number of persons.
• The distance between A and B is 4 persons, so $|P_A - P_B| = 5$. The distance between B and C is 7 persons, so $|P_B - P_C| = 8$.
• Since $P_C = 12$, $P_B$ must be either $12 - 8 = 4$ or $12 + 8 = 20$.
• Case 1: If $P_B = 4$, then $P_A$ is either $4 - 5 = -1$ (impossible) or $4 + 5 = 9$. If $P_A = 9$, then $N = P_A + 13 = 9 + 13 = 22$.
• Case 2: If $P_B = 20$, then $P_A$ is either $20 - 5 = 15$ or $20 + 5 = 25$. If $P_A = 15$, $N = 15 + 13 = 28$. If $P_A = 25$, $N = 25 + 13 = 38$.
• The possible values for N are 22, 28, and 38. The minimum is 22.
Why the other options are wrong
- (b) 28
- 28 is a possible total number of persons (when A is at 15, B at 20, C at 12), but it is not the minimum.
- (c) 32
- 32 is not a valid total number of persons for any arrangement satisfying the given distance constraints.
- (d) 38
- 38 is a possible total number of persons (when A is at 25, B at 20, C at 12), but it is not the minimum.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.