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 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
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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.
Updated On: Jul 16, 2026
- 11360
- 11520
- 21530
- none of these
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Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Fix the seating structure.
The family has 1 + 5 + 8 = 14 members in total, seated in a row of 14 seats. The 8 grandchildren occupy the 4 seats at each end, that is seats 1 to 4 and seats 11 to 14. This leaves the middle 6 seats, positions 5 to 10, for the grandfather and the 5 sons and daughters.
The 8 grandchildren can be arranged in these 8 end seats in \(8!\) ways.
Step 2: Place the grandfather so he has no grandchild beside him.
Within the middle block (positions 5 to 10), the grandfather cannot sit at position 5 (next to the grandchild at position 4) or position 10 (next to the grandchild at position 11), because those seats are adjacent to the grandchildren's block. So the grandfather has only 4 valid seats to choose from, positions 6, 7, 8 or 9.
The remaining 5 middle seats are filled by the 5 sons and daughters in \(5!\) ways.
Final Answer:
Total number of arrangements = \(8! \times 4 \times 5! = 40320 \times 4 \times 120 = 19{,}353{,}600\).
This number does not match 11360, 11520 or 21530, so the correct choice is "none of these". \[ \boxed{\text{Answer: none of these}} \]
The family has 1 + 5 + 8 = 14 members in total, seated in a row of 14 seats. The 8 grandchildren occupy the 4 seats at each end, that is seats 1 to 4 and seats 11 to 14. This leaves the middle 6 seats, positions 5 to 10, for the grandfather and the 5 sons and daughters.
The 8 grandchildren can be arranged in these 8 end seats in \(8!\) ways.
Step 2: Place the grandfather so he has no grandchild beside him.
Within the middle block (positions 5 to 10), the grandfather cannot sit at position 5 (next to the grandchild at position 4) or position 10 (next to the grandchild at position 11), because those seats are adjacent to the grandchildren's block. So the grandfather has only 4 valid seats to choose from, positions 6, 7, 8 or 9.
The remaining 5 middle seats are filled by the 5 sons and daughters in \(5!\) ways.
Final Answer:
Total number of arrangements = \(8! \times 4 \times 5! = 40320 \times 4 \times 120 = 19{,}353{,}600\).
This number does not match 11360, 11520 or 21530, so the correct choice is "none of these". \[ \boxed{\text{Answer: none of these}} \]
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