Proof
An element g 
Sn is even (odd),
if and only if the product g(1,2) is odd (even).
Hence the map g -> g(1,2) defines
a bijection between the even and the odd elements of Sn.
But then precisely half of the n! elements of Sn
are even.
An element g Sn is even (odd),
if and only if the product g(1,2) is odd (even).
Hence the map g -> g(1,2) defines
a bijection between the even and the odd elements of Sn.
But then precisely half of the n! elements of Sn
are even.