If $^{n}P_{4} = 24 \times ^{n}C_{5}$,then the value of $n$ is

If $\frac{1}{6!} + \frac{1}{7!} = \frac{x}{8!}$,find $x$.

An eight-digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is: (in $(7!)$)

The number of ordered pairs $(m, n)$,where $m, n \in \{1, 2, 3, \ldots, 50\}$,such that $6^m + 9^n$ is a multiple of $5$ is

The number of positive integral solutions of the equation $xyz = 3000$ is

The total number of $5$-digit telephone numbers that can be formed such that at least one digit is repeated is .... (The first digit cannot be $0$).