Question

A family consists of a grand father, $5$ sons and daughters and $8$ grand children. They are to be seated in a row for dinner. The grand children wish to occupy the $4$ seats at each end and the grandfather refuses to have a grand child on either side of him. Let "k" be the number of ways the family be made to sit.Find the sum of digits of k?

Answer 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
$=8!\times 4\times 120$
$=19353600$
Sum of digit is 27.