(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?

For every integer $n$ from $0$ to $6$, we have $3$ identical weights with weight $2^n$ grams. How many total ways are there to form a total weight of $263$ grams using only these weights?

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 $$\frac{(m+n)!}{(m+n)^{m+n}}<\frac{m!}{m^m}\frac{n!}{n^n}$$

How many different ways are there to make a payment of $n$ dollars using any number of $\$1$ and $\$2$ bills?

How many $4$-digit integers are there whose sum of all digits equals $12$?

There are $10$ red, $10$ blue, and $10$ white balls. How many different ways are there to retrieve $16$ balls with all the three colors present.

Let $\alpha(n)$ be the number of ways to write a positive integer $n$ as the sum of $1$s and $2$s. Let $\beta(n)$ be the number of ways to write $n$ as a sum of several integers greater than $1$. Different orders are treated as different. Prove $\alpha(n)=\beta(n+2)$.

A deck of poker has three different colors each of which contains $10$ cards numbered from $1$ to $10$, respectively. In addition, there are two jokers both of which are numbered as $0$. A card with number $k$ is valued as $2^k$ points. How many different ways are there to draw several cards from this deck so that their total value equals $2004$?

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

Let $n$ be a positive integer. Find the number $a_n$ of polynomials $f(x)$ with coefficients in $\{0, 1, 2, 3\}$ such that $f(2)=n$.

Let $a_0$, $a_1$, $a_2$, $\cdots$ be an increasing sequence of non-negative integers such that every non-negative integer can be expressed uniquely in the form of $(a_i + 2a_j+4a_k)$ where $i$, $j$, and $k$ are not necessarily distinct. Determine $a_{1998}$.

Let $\mathbb{N}$ be the set containing all positive integers. Is it possible to partition $\mathbb{N}$ to more than one but still a finite number of arithmetic sequences with no two having the same common difference?

Let $p$ be an odd prime number. Find the number of subsets $\mathbb{A}$ of the set $\{1, 2, \cdots, 2p\}$ such that $\mathbb{A}$ has exactly $p$ elements and the sum of all elements in $\mathbb{A}$ is divisible by $p$.

Let $a$, $b$, $p$, and $q$ be fixed positive integers. If an $a\times b$ grid can be tiled using some $1\times p$ and $q\times 1$ pieces, show that either $a$ is divisible by $p$ or $b$ is divisible by $q$. Here, a $1\times k$ and $k\times 1$ grids are treated as different.

Show that $$\frac{1}{1-x}=1+x+x^2+x^3+x^4 + \cdots$$