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# A family consist of a grandfather, 5 sons and daughter 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. 21530

B. 8! × 360

C. 8! × 480

D. 8! × 240

### Solution(By Examveda Team)

Total no. of seats,
= 1 grandfather + 5 sons and daughters + 8 grandchildren
= 14
The grandchildren can occupy the 4 seats on either side of the table in 4! = 24 ways.
The grandfather can occupy a seat in (5 - 1) = 4 ways (4 gaps between 5 sons and daughter).
And, the remaining seats can be occupied in 5! = 120 ways (5 seat for sons and daughter).
Hence total number of required ways,
= 8! × 480

This Question Belongs to Arithmetic Ability >> Permutation And Combination

1. The total number of seats
= 1 grandfather + 5 sons and daughters + 8 grand children
= 14
The grand children to occupy 8 seats on either side of the table
= 8! ways
And grand father can occupy a seat in (5−1) ways = 4 ways (since 4 gaps between 5 sons and daughters)
and the remaining seat can be occupied in 5! ways
= 120 ways (5 seats for sons and daughters)
Hence, the total number of ways, By the principle of multiplication law
=8!×4×120
=19353600

2. how this 8! came ?

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