Combinations of sets raised to the power of a prime modulus: $\binom{p^{\alpha}-1}k \equiv (-1)^k \pmod p$
Solution 1:
$${p^a-1\choose k}={(p^a-1)(p^a-2)\cdot\dots\cdot(p^a-k)\over(1)(2)\cdot\dots\cdot(k)}\equiv{(-1)(-2)\cdot\dots\cdot(-k)\over(1)(2)\cdot\dots\cdot(k)}=(-1)^k$$ where the congruence is modulo $p$.