In how many ways can the letters of the word '$MATHEMATICS$' be arranged,when all vowels come together?

The number of ways in which $5$ boys and $3$ girls can be seated in a row so that each girl is between two boys is:

An urn contains $5$ red marbles,$4$ black marbles,and $3$ white marbles. The number of ways in which $4$ marbles can be drawn so that at most three of them are red is:

$^{80}C_{40}$ is not divisible by -

From a group of $7$ men and $6$ women,five persons are to be selected to form a committee so that at least $3$ men are there on the committee. In how many ways can it be done?

There are $m$ men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84,$ then the value of $m$ is