The number of four-lettered words that can be formed from the letters of the word $MAYANK$ such that both $A$'s are included but never together,is equal to:

Three schools send $2, 4,$ and $6$ students,respectively,to a summer camp. The $12$ students must be accommodated in $6$ rooms numbered $1, 2, 3, 4, 5, 6$ in such a way that each room has exactly $2$ students and both are from the same school. The number of ways the students can be accommodated in the rooms is

The number of distinct primes dividing $12! + 13! + 14!$ is

Assuming the balls to be identical except for their color,the number of ways in which one or more balls can be selected from $10$ white,$9$ green,and $7$ black balls is:

Let $S$ be the set of all seven-digit numbers that can be formed using the digits $0, 1$ and $2$. For example,$2210222$ is in $S$,but $0210222$ is not in $S$. Then the number of elements $x$ in $S$ such that at least one of the digits $0$ and $1$ appears exactly twice in $x$,is equal to $....$

The sum of all $4$-digit numbers that can be formed using the digits $2, 3, 4, 5, 6$ without repetition is: