# 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

**Answer: Option C **

__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

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Related Questions on Permutation and Combination

A. 3! 4! 8! 4!

B. 3! 8!

C. 4! 4!

D. 8! 4! 4!

A. 7560,60,1680

B. 7890,120,650

C. 7650,200,4444

D. None of these

A. 8 × 9!

B. 8 × 8!

C. 7 × 9!

D. 9 × 8!

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

how this 8! came ?