For n > 1, prove that the number of even permutations of [n] equals the number of odd permutations of [n].
Let $\displaystyle \pi$ be a permutation of [n]. Define another permutation $\displaystyle \sigma$ by switching $\displaystyle \pi(1)$ and $\displaystyle \pi(2)$; i.e., define
$\displaystyle \sigma(1) = \pi(2)$
$\displaystyle \sigma(2) = \pi(1)$
$\displaystyle \sigma(j) = \pi(j)$ for $\displaystyle j > 2$.
Since $\displaystyle \pi$ and $\displaystyle \sigma$ differ only by a transposition, they have opposite parity, i.e. one is odd and one is even. So we have established a bijection between the even and odd permutations, and there are equal numbers of each.