Question

A family consists of a grandfather, $m$ sons and daughters and $2n$ grandchildren. They are to be seated in a row for dinner. The grandchildren wish to occupy the $n$ seats at each end and the grandfather refuses to have a grandchildren on either side of him. In how many ways can the family be made to sit?

• A. $\left(2n\right)!m!(m-1)$
• B. $\left(2n\right)!m!(m-2)$
• C. $\left(n\right)!m!(m-1)$
• D. $\left(2n\right)!(m+1)!$

Answer 1

The correct option is A $\left(2n\right)!m!(m-1)$

The number of ways of arranging the grand children at each end's $\left(2n\right)!$

Sons and daughter can be occupies by $m!$

While grandfather will occupy the remaining seat in middle i.e. $(m-1).$

Note! After grandfather only we shall arrange son and daughter

$\therefore$ Total ways$=2n!\times m!(m-1)$

Hence the answer is $\left(2n\right)!m!(m-1).$