Binomial Coefficient involving Power of Prime

Lemma
Let $$p$$ be a prime number, and let $$k \in \Z$$.

Then $$\binom {p^n k} {p^n} \equiv k \left({\bmod\, p}\right)$$, where $$\binom {p^n k} {p^n}$$ is a binomial coefficient.

Proof
From Prime Power of Sum Modulo Prime we have:
 * $$(1) \qquad \left({a + b}\right)^{p^n} \equiv \left({a^{p^n} + b^{p^n}}\right) \left({\bmod\, p}\right)$$

We can write this:
 * $$\left({a + b}\right)^{p^n k} = \left({\left({a + b}\right)^{p^n}}\right)^k$$

By $$(1)$$ and Congruence of Powers, we therefore have:
 * $$\left({a + b}\right)^{p^n k} \equiv \left({a^{p^n} + b^{p^n}}\right)^k \left({\bmod\, p}\right)$$

The coefficient $$\binom {p^n k} {p^n}$$ is the binomial coefficient of $$b^{p^n}$$ in $$\left({a + b}\right)^{p^n k} = \left({\left({a + b}\right)^{p^n}}\right)^k$$.

Expanding this using the Binomial Theorem, we find that the coefficient of $$b^{p^n}$$ is $$k$$.

So $$\binom {p^n k} {p^n} \equiv k \left({\bmod\, p}\right)$$.

Comment
This lemma is used in the proof of the Sylow Theorems.