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Question
Five men, two women, and a child sit around a table. Find the number of arrangements where the child is seated between two men.
Solution
Five men, two women, and a child sit around a table.
Here, the child is seated between 2 men.
Since there are 5 men, such a group can be formed in 5P2
= `(5!)/(3!)`
= 5 × 4 = 20 ways
Thus, there are 20 ways in which the child can be seated between 2 men.
We consider the 2 men and the child between as one unit.
Also, we have 3 more men and 2 women.
Thus, we have 1 + 3 + 2 = 6 persons.
These 6 persons can be arranged around a table in (6 – 1)! = 5! – ways.
∴ Total number of required arrangements
= 20 × 5!
= 20 × 120
= 2400
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