Find the number of different signals that can be generated by arranging at least $2$ flags in order (one below the other) on a vertical staff,if five different flags are available.

For which value of $n \in N$,does $n!$ have $13$ trailing zeros?

Five-digit numbers are formed using the digits $1, 2, 3, 4, 5, 6$,and $8$. What is the probability that they have even digits at both the ends?

How many natural numbers $n$ are there such that $n! + 10$ is a perfect square?

In how many ways can $3$ letters be posted in $4$ letter boxes,if all the letters are not posted in the same letter box?

There are $(n + 1)$ white and $(n + 1)$ black balls,with each set numbered $1$ to $n + 1$. The number of ways in which the balls can be arranged in a row so that no two adjacent balls are of the same colour is