The number of arrangements that can be formed from the letters $a, b, c, d, e, f$ taken $3$ at a time without repetition,such that each arrangement contains at least one vowel,is

The sum of the digits in the unit place of all numbers formed with the help of $3, 4, 5, 6$ taken all at a time is

How many $6$-digit numbers can be formed using the digits $1, 2, 3,$ and $4$ such that the number contains exactly two pairs of identical digits?

The number of ways in which ten candidates $A_1, A_2, \dots, A_{10}$ can be ranked such that $A_1$ is always above $A_{10}$ is:

The sum of all $4$-digit numbers that can be formed by using the digits $2, 4, 6, 8$ (repetition of digits not allowed) is

Eight chairs are numbered $1$ to $8$. Two women and three men wish to occupy one chair each. First,the women choose the chairs from amongst the chairs marked $1$ to $4$,and then the men select the chairs from amongst the remaining. The number of possible arrangements is