Find the remainder when $1\times 2 + 2\times 3 + 3\times 4 + \cdots + 2018\times 2019$ is divided by $2020$.

Assuming a small packet of mm’s can contain anywhere from $20$ to $40$ mm’s in $6$ different colours. How many different mm packets are possible?

Let $n > k$ be two positive integers. Simplify the following expression $$\binom{n}{k} + 2\binom{n-1}{k} + 3\binom{n-2}{k} + \cdots+ (n-k+1)\binom{k}{k}$$

Compute: $1\times 2\times 3 + 2\times 3\times 4 + \cdots + 18\times 19\times 20$.

(Hockey Sticker Identity) Show that for any positive integer $n \ge k$, the following relationship holds: $$\binom{k}{k} +\binom{k+1}{k} + \binom{k+2}{k} + \cdots + \binom{n}{k} = \binom{n+1}{k+1} $$

Show that $$\frac{1}{(1-x)^n}=\sum_{k=0}^{\infty}\binom{n-1+k}{n-1}x^k$$

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 $n$ 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 $n$?