Calculating minimum queue length with positional constraints

To minimize the total queue length, we must maximize the overlap of the specified intervals. This means placing the components inside each other.

There are 11 persons ahead of C. This strictly fixes C at position 12 from the front.

To minimize people added to the back, we want B and A to be positioned ahead of C if possible. There are 7 persons between B and C. Placing B ahead of C: Position of B = 12 - 7 (gap) - 1 (for B itself) = 4.

There are 4 persons between A and B. Can we place A ahead of B? 4 - 4 (gap) - 1 = -1 (Impossible index). Therefore, A must be placed behind B. Position of A = 4 + 4 (gap) + 1 (for A itself) = 9.

Verify the configuration:

Now calculate the total queue length. There are exactly 13 persons behind A. Total = A's position + 13 = 9 + 13 = 22.

Answer: (a).