The number of ways in which $6$ distinct rings can be worn on the $4$ fingers of one hand is:

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

The number of words that can be formed out of the letters of the word $ARTICLE$ so that the vowels occupy even places is

The number of ways in which $5$ boys and $3$ girls can be seated on a round table if a particular boy $B_1$ and a particular girl $G_1$ never sit adjacent to each other,is

Find $n$ if ${ }^{n} P_{4}=18 \cdot{ }^{n-1} P_{2}$.

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.