Calculate $\binom{n}{0},\binom{n}{1}$ and $\binom{n}{2}$.
There is only one subset with 0 elements (the empty set), so $\binom{n}{0}=1$.

There are as many subsets with 1 element (called singletons) as there are elements, so $\binom{n}{1}=n$.

To obtain a subset with 2 elements, we start by choosing an element $a$ ($n$ possibilities) and then an element $b \neq a$ ($n-1$ possibilities). However, we have counted each set $\{a, b\}$ and $\{b, a\}$ twice. Consequently, $\binom{n}{2}=\frac{n(n-1)}{2}$.

## Classical Identities

In this section, we derive some classical identities for binomial coefficients using the principle of "double-counting": we consider a set and count its number of elements in two different ways. It will then suffice to equate the two values obtained to get (without further calculation) a new identity.