In how many ways can 3 ladies and 3 gentlemen be seated around a round table so that any two and only two of the ladies sit together?

Answer

72

Explanation

The number of selection of two ladies to sit together is `.^(3)C_(2)`.
Let the two seats occupied by these two (selected ) ladies be numbered as 1 and 2 and remaining seats by 3,4,5, and 6. The `3^(rd)` lady cannot occupy seat number 3 and 6, so there are only two choices (viz., numbers 4 and 5) left with the `3^(rd)` lady who can make selection of seats in `.^(2)C_(1)` ways.
Now, the two selected ladies (in seat numbers 1 and 2) can be permuted (i.e., change their seats ) in 2! ways and 3 gentlemen can be permuted on remaining 3 seats in 3! ways.
Hence, by product rule, total number of ways is `.^(3)C_(2)xx .^(2)C_(1)xx2!xx3!=72`.