How many permutations can be made by using all the letters of the word $MATHEMAGICA$?

Let $T_n$ denote the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides. If $T_{n + 1} - T_n = 21$,then $n$ equals:

The number of $4$-digit numbers that can be formed from the digits $0, 1, 2, 3, 4, 5, 6, 7$ such that each number contains the digit $1$ is:

The number of ways in which $6$ men and $5$ women can dine at a round table if no two women are to sit together is given by

Ten students are participating in a race. In how many ways can the first $3$ prizes be won?

There are $6$ men and $8$ women in a club in which a committee of $5$ people has to be formed. What will be the number of ways to select $5$ people if no man is selected?