What n choose k counts

The binomial coefficient inom{n}{k} counts how many subsets of size $k$ can be formed from $n$ distinct items when order is ignored.

Why symmetry helps

Choosing the $k$ selected items is equivalent to choosing the $n-k$ items left behind, so inom{n}{k} = inom{n}{n-k}.

Where it appears

Combinations appear in probability, Pascal’s triangle, committee selection, lottery counting, and the expansion of $(a+b)^n$.