The binomial coefficient

$$\binom nk$$

counts the ways to choose $k$ objects from $n$ without regard to order.

These coefficients appear naturally when expanding a power of a sum. In

$$(a+b)^n,$$

a term $a^{n-k}b^k$ is produced by choosing which $k$ of the $n$ factors contribute $b$. Therefore

$$(a+b)^n=\sum_{k=0}^n \binom nk a^{n-k}b^k.$$

This is the binomial theorem.

The coefficients satisfy useful identities such as

$$\binom nk=\binom n{n-k}$$

and Pascal's identity

$$\binom nk=\binom{n-1}{k-1}+\binom{n-1}{k}.$$

Such identities can be derived algebraically or proved by counting the same set of objects in two different ways.