If numbers $1, 2, ..., n$ are ordered in a sequence $a_1, a_2, ..., a_n$ and if $n$ is odd, prove that $ (a_1-1)(a_2-2)...(a_n-n)$ is even.

Question

If numbers $1, 2, ..., n$ are ordered in a sequence $a_1, a_2, ..., a_n$ and if $n$ is odd, prove that $ (a_1-1)(a_2-2)...(a_n-n)$ is even.

I tried to write that product as a sum, I took 2 from the second factor and anything from other factors for the first summand and the second summand you get when you take a_2 from the second factor. First is obviously even and can someone prove that the other one also is divided by 2.

Can anyone help?

Any ideas?