Lucas' theorem

Statement
Suppose $$a$$ and $$b$$ are nonnegative integers and $$p$$ is a prime number. Suppose $$a = a_0 + a_1p + a_2p^2 + \dots + a_rp^r$$ and $$b = b_0 + b_1p + b_2p^2 + \dots + b_sp^s$$ are the expressions of $$a$$ and $$b$$ in base $$p$$, so that each $$a_i, b_j$$ is in the set $$\{ 0,1,2,\dots,p-1 \}$$. if $$r < s$$, define $$a_i = 0$$ for $$r < i \le s$$. Then, we have:

$$\binom{a}{b} \equiv \prod_{i=0}^s \binom{a_i}{b_i} \pmod p$$

By convention, $$\binom{x}{y} = 0$$ if $$x = 0$$ or if $$y > x$$.

In particular, we have the following: If $$a$$ is a power of $$p$$ and $$b < a$$, then $$\binom{a}{b}$$ is relatively prime to $$p$$. For more on this special case and alternative proofs of it, see Lucas' theorem prime power case.