1. Tardigrade
  2. Question
  3. Mathematics
  4. A family consists of 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 grand child on either side of him. The number of ways in which the family can be made to sit is

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Q. A family consists of 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 grand child on either side of him. The number of ways in which the family can be made to sit is

Permutations and Combinations

Solution:

The total number of seats
$= 1$ grand father $+ 5$ sons and daughters$+ 8$ grand children
$= 14$
The grand children wish to occupy the $4$ seats on either side of the table $=4 !$ ways
$=24$ 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 seats can be occupied in $5 !$ ways
$=120$ ways ( $5$ seats for sons and daughters)
Hence, required number of ways
$=24 \times 4 \times 120=11520$