For a natural number $n$,the inequality $2^n(n - 1)! < n^n$ holds true if:

The sum of all the four-digit numbers that can be formed using all the digits $2, 1, 2, 3$ is equal to $.......$.

The number of times the digit $3$ will be written when listing the integers from $1$ to $1000$ is

The number of seven-digit integers with the sum of the digits equal to $10$ and formed by using the digits $1, 2,$ and $3$ only is:

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

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: