The number of ways in which $6$ men and $5$ women can dine at a round table if no two women are to sit together is given by

Find the number of ways in which two Americans,two British,one Chinese,one Dutch,and one Egyptian can sit at a round table so that persons of the same nationality are separated.

The number of ways in which $6$ men and $4$ women can be seated around a table so that a particular man and a particular woman never sit adjacent to each other is

$A$ couple can sit with $6$ guests around a circular table. In how many ways can they sit if the couple sits in consecutive seats?

In how many ways can $8$ beads of different colors be strung into a necklace?

If $5$ red roses and $5$ white roses of different sizes are used in preparing a garland,the probability that red and white roses come alternately is