A joint family in a town has a grandmother, five children and two grandchildren. In how many ways can the seating arrangement be made such that the grandchildren occupy both the seats at the end and the grandmother sits with her children on either side of her?
Question
A joint family in a town has a grandmother, five children and two grandchildren. In how many ways can the seating arrangement be made such that the grandchildren occupy both the seats at the end and the grandmother sits with her children on either side of her?
Solution
The seating arrangement can be thought of as follows:
- The two grandchildren will occupy the two ends. They can be arranged in 2! = 2 ways.
- The grandmother will sit in the middle of her five children. This group (grandmother + 5 children) can be thought of as a single entity, and they can be arranged in 5! = 120 ways.
- Within this group, the grandmother can sit on either side of the group of five children, so there are 2 ways to arrange the grandmother and the children.
Therefore, the total number of ways the seating arrangement can be made is 2 (ways to arrange the grandchildren) * 120 (ways to arrange the children and the grandmother as a group) * 2 (ways to arrange the grandmother within the group) = 480 ways.
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