Lucas' Theorem

For a prime number \(p\), Let
$$m=m_{k}p^{k}+m_{k-1}p^{k-1}+ \ldots + m_{0}p^{0},$$
$$n=n_{k}p^{k}+n_{k-1}p^{k-1}+ \ldots + n_{0}p^{0}$$
where \(0 \leq m_{i}, n_{i} < p\)

Lucas' Theroem states that:
$$\dbinom{m}{n} \equiv \displaystyle{\prod_{i=0}^{k}} \dbinom{m_{i}}{n_{i}} \mod p$$