Correct Option
To determine the minimum number of persons in the queue, an arrangement that maximizes overlap among the specified groups must be considered. Let's assume the order of persons in the queue is B, then A, then C, from front to back.
- There are 4 persons between A and B. This implies the sequence: B (4 persons) A.
- There are 7 persons between B and C. Given the B, A, C order, these 7 persons include the 4 persons between B and A, and A itself. Thus, the remaining persons between A and C must be 7−4−1=2 persons. This establishes the sequence: B (4 persons) A (2 persons) C.
- There are 11 persons ahead of C. In our established sequence, the persons ahead of C are the 2 persons between A and C, A itself, the 4 persons between B and A, and B itself. This totals 2+1+4+1=8 persons. Therefore, the number of persons ahead of B must be 11−8=3 persons. The sequence becomes: (3 persons) B (4 persons) A (2 persons) C.
- There are 13 persons behind A. In our sequence, the persons behind A are the 2 persons between A and C, and C itself. This totals 2+1=3 persons. Therefore, the number of persons behind C must be 13−3=10 persons. The final sequence is: (3 persons) B (4 persons) A (2 persons) C (10 persons).
Verifying all conditions with this arrangement:
- Persons between A and B: 4. (Matches)
- Persons between B and C: (4 persons) + A + (2 persons) = 4+1+2=7. (Matches)
- Persons ahead of C: (3 persons) + B + (4 persons) + A + (2 persons) = 3+1+4+1+2=11. (Matches)
- Persons behind A: (2 persons) + C + (10 persons) = 2+1+10=13. (Matches)
The total minimum number of persons in the queue is the sum of all segments and individuals: 3+1+4+1+2+1+10=22.
Incorrect Options
Options 2) 28, 3) 32, and 4) 38 represent higher numbers of persons than the calculated minimum. These values would result from scenarios involving less overlap between the groups (e.g., A, B, C in a linear, non-overlapping fashion, or different relative orders that increase the total count) or scenarios where not all conditions are simultaneously met for a minimum configuration. The detailed calculation demonstrates that 22 is the lowest possible number satisfying all given constraints.