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

The number of ways of arranging all the letters of the word "$COMBINATIONS$" around a circle so that no two vowels come together is

In how many ways can $6$ men and $5$ women be seated around a circular table if no two women are to be seated together?

The number of ways of arranging $9$ men and $5$ women around a circular table so that no two women come together are

$A$ round table conference is to be held amongst $20$ countries. If two particular delegates wish to sit together,then in how many ways can such arrangements be done?

Let $m$ denote the number of ways in which $5$ boys and $5$ girls can be arranged in a line alternately,and $n$ denote the number of ways in which $5$ boys and $5$ girls can be arranged in a circle so that no two boys are together. If $m = kn$,then the value of $k$ is