The number of $5$-digit numbers that are not divisible by $5$ and consist of different odd digits is:

An envelope has space for at most $3$ stamps. If you are given three stamps of denomination $1$ and three stamps of denomination $a$ (where $a > 1$),what is the least positive integer for which there is no possible stamp value?

Let $n = 1! + 4! + 7! + \ldots + 400!$. Then the ten's digit of $n$ is

Let the eleven letters $A, B, ....., K$ denote an arbitrary permutation of the integers $(1, 2, ....., 11)$. Then,what is the nature of the product $(A - 1)(B - 2)(C - 3) ..... (K - 11)$?

$A$ 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 invited,is:

The number of $5$-digit numbers that can be formed using the digits $1, 2, 3, 4, 5, 6$ such that the number must include both $1$ and $2$ is: