There are $15$ buses running between Delhi and Mumbai. In how many ways can a man go to Mumbai and return by a different bus?

If the letters of the word $MOTHER$ are permuted and all the words so formed (with or without meaning) are listed as in a dictionary,then the position of the word $MOTHER$ is

In how many ways can $21$ books on Sanskrit and $19$ books on French be placed in a row on a shelf such that two books on French may not be together?

There are $5$ letters and $5$ directed envelopes. Find the number of ways in which the letters can be put into the envelopes so that all are not put in directed envelopes?

$m$ men and $n$ women are to be seated in a row so that no two women sit together. If $m > n$,then the number of ways in which they can be seated is

If a man and his wife enter a bus in which five seats are vacant,then the number of different ways in which they can be seated is