Let $n = 1! + 4! + 7! + \ldots + 400!$. Then the ten's digit of $n$ is

Four notes of Rs. $100$ and one note each of Rs. $1$,Rs. $2$,Rs. $5$,Rs. $20$,and Rs. $50$ are to be distributed among $3$ children such that each child receives at least one note of Rs. $100$. In how many ways can this distribution be done?

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

Numbers between $1$ and $10,000$ are formed using the digits $2$ and $3$ exactly once and the digit $4$ twice. If the numbers thus formed are arranged in increasing order and $x, y$ represent the ranks of $4324$ and $324$ respectively,then $x-y=$

If the number of circular permutations of $9$ distinct things taken $5$ at a time is $n_1$ and the number of linear permutations of $8$ distinct things taken $4$ at a time is $n_2$,then $\frac{n_1}{n_2}=$

Let $x$ and $y$ be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25$. Then,the number of ways of choosing $x$ and $y$ such that $x + y$ is divisible by $5$ is $.........$.