In how many ways can $4$ married couples seat themselves around circular table if there are no restrictions?

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Solution

Arrange one person at first. Since it is a circular table, it does not make any difference where he sits.
Other seats are then distinguishable relative to the occupied seat.
Hence the remaining $7$ can be arranged in $7!$ ways.
Total number of ways $= 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5, 040 ‬$ ways.

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In how many ways can 4 married couples seat themselves around circular table if there are no restrictions?

Arrange one person at first. Since it is a circular table, it does not make any difference where he sits. Other seats are then distinguishable relative to the occupied seat. Hence the remaining 7 can be arranged in 7! ways. Total number of ways =7!=7×6×5×4×3×2×1=5,040‬ ways.

In how many ways can $4$ married couples seat themselves around circular table if there are no restrictions?