Question

A family consists of a grandfather, 5 sons and daughters and 8 grandchildren. They are to be seated in a row for dinner. The grandchildren wish to occupy the 4 seats at each end and the grandfather refuses to have a grandchild on either side of him. The number of ways in which the family can be made to sit is

• A. 11360
• B. 11520
• C. 21530
• D. none of these

Hint:

Seat the 8 grandchildren in the end blocks first (8! ways), then find how many of the 6 middle seats the grandfather can safely occupy without sitting next to a grandchild.

Answer 1

The Correct Option is D

Instead of directly counting the grandfather's valid seats, we can count all arrangements of the middle block and subtract the invalid ones where the grandfather sits at an end of that block.

1. All arrangements of the middle block: the grandfather plus 5 sons and daughters, 6 people in 6 seats, can be arranged in \(6! = 720\) ways with no restriction.
2. Invalid arrangements: these are the ones where the grandfather sits at position 5 or position 10 of the row (the two ends of the middle block, each adjacent to a grandchild). Fixing the grandfather at either of these 2 spots, the remaining 5 people fill the other 5 seats in \(5! = 120\) ways, giving 2 x 120 = 240 invalid arrangements.
3. Valid middle block arrangements: 720 - 240 = 480, which equals 4 x 5! from the direct method, confirming the two approaches agree.

Multiplying by the \(8!\) ways to seat the grandchildren at the two ends gives \(8! \times 480 = 19{,}353{,}600\) total arrangements, a number that matches none of options A, B or C, so the correct answer is "none of these" (option D).