If $(n+2)! = 2550(n!)$,find $n$.

In how many ways can $15$ different books be divided equally among $5$ sets or groups?

If ${}^n{P_5} = 20 \cdot {}^n{P_3}$,then $n = $

Let $n > 2$ be an integer. Suppose that there are $n$ Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track. Further,each pair of nearest stations is connected by a blue line,whereas all remaining pairs of stations are connected by a red line. If the number of red lines is $99$ times the number of blue lines,then the value of $n$ is

The number of four-digit natural numbers which contain exactly two distinct digits is:

The number of ways in which four letters of the word '$MATHEMATICS$' can be arranged is given by