Which one of the following is possible ?
- (a) D is 20 years old
- (b) F is 18 years old ✓ UPSC's answer
- (c) F is 19 years old
- (d) F is 20 years old
Why the answer is (b)
• The ages of the six cousins are distinct integers ranging from 17 to 22, so the set of ages is {17, 18, 19, 20, 21, 22}.
• E is the oldest, so E = 22. The youngest is 17, so one of A, B, C, D, F is 17.
• A is one year older than C (A = C + 1), and A is older than B, and C is older than D. This implies the age order A > C > D and A > B.
• F is between B and D in age, meaning either B < F < D or D < F < B.
• Let's test the possible values for C. Since A = C + 1 and A ≤ 21 (since E=22 is max), C can be 17, 18, 19, or 20.
• If C = 17, A = 18. Then D < 17 is impossible since 17 is the minimum age. So C cannot be 17.
• If C = 18, A = 19. D < 18, so D must be 17. B < A (19) and B is not 17 (D is 17), so B can be 20 or 21. F is between B and D (17). If B=20, F is 18 or 19. But C=18, so F cannot be 18. If F=19, A=19, conflict. If B=21, F is 18, 19, or 20. C=18, A=19. F cannot be 18 or 19. So F=20. Remaining age for the last person is 21? No, ages used: E=22, A=19, C=18, D=17, F=20, B=21. Check constraints: A(19)>B(21)? No, 19 is not > 21. So this case fails because A must be older than B.
• If C = 19, A = 20. D < 19, so D is 17 or 18. B < 20. F is between B and D.
• Case 1: D = 17. B can be 18, 21, 22? No, E=22. B can be 18 or 21. If B=18, F is between 18 and 17? No integer between. If B=21, F is between 21 and 17, so F is 18, 19, or 20. A=20, C=19. F cannot be 19 or 20. So F=18. Ages: E=22, A=20, C=19, D=17, F=18, B=21. Check: A(20)>B(21)? No. Fails.
• Case 2: D = 18. B < 20. B can be 17, 21. If B=17, F between 17 and 18? No integer. If B=21, F between 21 and 18, so F is 19 or 20. A=20, C=19. F cannot be 19 or 20. Fails.
• If C = 20, A = 21. D < 20, so D is 17, 18, or 19. B < 21. F is between B and D.
• Let's check option (b) F = 18. If F=18, then B and D must be on either side of 18. So one is <18 and one is >18. The only age <18 is 17. So one of B or D is 17. The other is >18 (19, 20, 21, 22). But D < C=20, so D can be 17, 18, 19. B < A=21, so B can be 17, 18, 19, 20.
• If D=17, then B > 18. B can be 19 or 20. F=18 is between 17 and B. This works. Let's assign: E=22, A=21, C=20, D=17, F=18. Remaining age for B is 19. Check: A(21) > B(19) Yes. C(20) > D(17) Yes. F(18) between B(19) and D(17) Yes. All ages distinct: 22, 21, 20, 17, 18, 19. This is a valid arrangement.
• Therefore, F can be 18 years old.
Why the other options are wrong
- (a) D is 20 years old
- If D is 20, then C must be older than 20, so C is 21 or 22. If C=21, A=22, but E=22, conflict. If C=22, A=23, impossible. Thus D cannot be 20.
- (c) F is 19 years old
- If F is 19, testing valid age assignments shows that no combination satisfies all constraints (A=C+1, A>B, C>D, F between B and D) without age conflicts or violating the 'older than' conditions.
- (d) F is 20 years old
- If F is 20, similar to option (c), no valid assignment of the remaining ages to A, B, C, D, E satisfies all the given relational constraints simultaneously.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2015, held on 23 August 2015. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.