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:

If there are $6$ periods in each working day of a school,in how many ways can one arrange $5$ subjects such that each subject is allowed at least $1$ period?

The number of ways of dividing $52$ cards amongst four players so that three players have $17$ cards each and the fourth player has just one card,is

$A$ man $X$ has $7$ friends,$4$ of them are ladies and $3$ are men. His wife $Y$ also has $7$ friends,$3$ of them are ladies and $4$ are men. Assume $X$ and $Y$ have no common friends. The total number of ways in which $X$ and $Y$ together can throw a party inviting $3$ ladies and $3$ men,such that $3$ friends of each of $X$ and $Y$ are in this party,is:

$A$ committee of $4$ persons is to be formed from $2$ ladies,$2$ old men,and $4$ young men such that it includes at least $1$ lady,at least $1$ old man,and at most $2$ young men. The total number of ways in which this committee can be formed is:

The number of diagonals in a polygon of $m$ sides is