In how many ways can $4$ beads be selected and arranged in a necklace from $6$ distinct beads?

$8$ teachers and $4$ students are sitting around a circular table at random. Then,the probability that no two students sit together is:

The number of ways in which $5$ boys and $4$ girls can sit around a circular table such that no two girls sit together is:

$A$ 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?

If $n$ persons are seated around a circular table,what is the ratio of the unfavorable probability to the favorable probability that two specific persons sit together?

If $3$ sisters and $8$ other girls are together playing a game,then the number of ways in which all the girls are seated around a circle such that the three sisters are not seated together,is