In how many different ways can $4$ married couples seat themselves around a circular table if (i) Spouses sit opposite each other? (ii) Men and women alternate?

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Solution

$4$ Married couples That is $8$ people
Second women can be placed in $6$ ways and her husband in $1$ way.
Third women can be placed in $4$ ways and her husband in $1$ way.
Fourth women can be placed in $2$ ways and her husband in $1$ way.
Total number of ways $6 \times 4 \times 2 = 48$ Ways.
Arranging Men
Number of ways $= 3! = 6$ Ways
Remaining $4$ places $4$ people $= 4! = 24$ Ways
Total ways of arranging $= 24 \times 6 = 144$ Ways

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In how many different ways can 4 married couples seat themselves around a circular table if (i) Spouses sit opposite each other? (ii) Men and women alternate?

In how many different ways can $4$ married couples seat themselves around a circular table if (i) Spouses sit opposite each other? (ii) Men and women alternate?