(Generalized binomial expansion) If $a$, $b$, and $r$ are some real or complex numbers, then $$(a+b)^r = \sum_{k=0}^{\infty}\binom{r}{k}a^{r-k}b^k$$

Here, the following definition still holds when $r$ is a real or complex number: $$\binom{r}{k}=\frac{r(r-1)\cdots(r-k+1)}{1\cdot 2\cdots k}$$

Show $$\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{m+k}{q}=(-1)^n\binom{m}{q-n}$$

Let $N$ be the value of the following expression. $$\sum_{k=0}^{n-1}\left(\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{k}\right)\left(\binom{n}{k+1}+\binom{n}{k+2}+\cdots+\binom{n}{n}\right)$$

Show $$N=\frac{n}{2}\binom{2n}{n}$$

Show that $$\sum_{k=1}^{n}(-1)^k\binom{n}{k}\left(1+\frac{1}{2}+\cdots+\frac{1}{k}\right)=-\frac{1}{n}$$

Prove $$\sum_{k=0}^{n}(-1)^k\frac{{n \choose k}}{\binom{m+k}{k}}=\frac{m}{m+n}$$

Evaluate the value of $$\sum_{m=0}^{2009}\sum_{n=0}^{m}\binom{2009}{m}\binom{m}{n}$$

Find the sum of all $n$ such that $$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+\cdots +\binom{n}{2018} = 0$$

Given randomly selected $5$ distinct positive integers not exceeding $90$, what is the expected average value of the fourth largest number?

Let $\mathbb{S}$ be a set of integers, $\max(\mathbb{S})$ be the largest element in $\mathbb{S}$, and $\mid\mathbb{S}\mid$ be the number of elements in $\mathbb{S}$. Find the number of non-empty set $\mathbb{S}\in\{1,2,\cdots,10\}$ satisfying $\max(\mathbb{S})\le\mid\mathbb{S}\mid + 2$.

Let $m$ and $n$ be positive integers. Show that $$\frac{(m+n)!}{(m+n)^{m+n}}<\frac{m!}{m^m}\frac{n!}{n^n}$$

Compute the value of $$\sum_{k=0}^{n}\frac{1}{2^k}\binom{n+k}{n}$$

Let $n$ be an even integer. Find the number of ways to select four distinct integers $a$, $b$, $c$, $d$ between $1$ and $n$, inclusive, satisfying $a+c=b+d$. Order of these four numbers does not matter.

How many $3\times 3$ matrices of non-negative integers are there such that the sum of every row and every column equals $n$?

Show that the limit of $f(n)=\left(1+\frac{1}{n}\right)^n$ exits when $n$ becomes infinitely large.