Consider the following two statements:
$I.$ If $n$ is a composite number,then $n$ divides $(n-1)!$.
$II.$ There are infinitely many natural numbers $n$ such that $n^3+2n^2+n$ divides $n!$.

$A$ set contains $2n + 1$ elements. The number of subsets of this set containing more than $n$ elements is equal to

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 $!$)

The number of all four-digit numbers which do not have four distinct digits is

For integers $n$ and $r$,let $\binom{n}{r} = \begin{cases} ^{n}C_{r}, & \text{if } n \geq r \geq 0 \\ 0, & \text{otherwise} \end{cases}$. The maximum value of $k$ for which the sum $\sum_{i=0}^{k}\binom{10}{i}\binom{15}{k-i} + \sum_{i=0}^{k+1}\binom{12}{i}\binom{13}{k+1-i}$ exists,is equal to ...... .