The number of numbers greater than $5000$ and less than $9000$ that are divisible by $3$, which can be formed using the digits $0, 1, 2, 5, 9$ with repetition allowed, is . . . . . . .

Out of $7$ consonants and $4$ vowels, words are formed each having $3$ consonants and $2$ vowels. The number of such words that can be formed is

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

From a group of $7$ batsmen and $6$ bowlers,$10$ players are to be chosen for a team,which should include at least $4$ batsmen and at least $4$ bowlers. One batsman and one bowler,who are the captain and vice-captain of the team respectively,must be included. The total number of ways such a selection can be made is:

Let $S = \{0, 1, 2, 3, \ldots, 100\}$. The number of ways of selecting $x, y \in S$ such that $x \neq y$ and $x + y = 100$ is

The number $(49^{2}-4)(49^{3}-49)$ is divisible by (in $!$)