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 the these

Answer 1

The Correct Option is D

The problem requires us to find the number of seating arrangements such that certain conditions are met. The family consists of 14 members in total: 1 grandfather (G), 5 sons and daughters (S), and 8 grandchildren (C). Let's denote the members as follows:

• G (grandfather)
• S₁, S₂, S₃, S₄, S₅ (sons and daughters)
• C₁, C₂, C₃, C₄, C₅, C₆, C₇, C₈ (grandchildren)

The seating conditions are:

1. The grandchildren must occupy the 4 seats at each end.
2. The grandfather cannot have a grandchild seated on either side of him.

We first arrange the 8 grandchildren. Since there are two sets of 4 seats at both ends, this can be done in:

Number of ways = 8!/(4!4!) = 70 ways

After arranging grandchildren, we have positions 5 through 10 available in the middle, where we can seat the grandfather and 5 sons/daughters. The grandfather can sit in one of the positions 6, 7, 8, or 9 to ensure no grandchild is seated beside him. After choosing one seat for the grandfather, the remaining sons/daughters can fill the remaining spots.
The calculation in each case:

• Choose 1 position for G out of 4: 4 ways
• Arrange 5 sons/daughters in the remaining 5 seats: 5! = 120 ways

Thus, the total arrangements are:

Total ways = 70 (for grandchildren) × 4 × 120 (for G, sons/daughters) = 33600 ways

Hence, the number of ways the family can be made to sit according to the conditions is 33600.
Therefore, the correct answer is none of these since 33600 does not match the given options.