Let $n > k$ be two positive integers. Simplify the following expression $$\binom{n}{k} + 2\binom{n-1}{k} + 3\binom{n-2}{k} + \cdots+ (n-k+1)\binom{k}{k}$$
Consider all $1000$-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. Find the arithmetic mean of all of these least elements.
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?
How many $3\times 3$ matrices of non-negative integers are there such that the sum of every row and every column equals $n$?