In how many ways can $5$ keys be put in a ring?

If $20$ beads of two different colors are to be arranged in a necklace such that they alternate,and there are $10$ beads of each color,then what is the number of ways to arrange them?

In how many ways can $10$ people be seated around a circular table such that no two arrangements have the same neighbors? (Clockwise and counter-clockwise arrangements are considered the same.)

$A$ 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?

$4$ gentlemen and $4$ ladies are to be seated at a round table. What is the probability that they sit alternately?

The number of ways in which $3$ boys and $4$ girls can be seated around a circular table if no two boys sit together is