Pigeonhole Principle Proof and Existence

Question

So, I’m going through a textbook on combinatorics, and I came across this exercise question.

Let $n$ be odd, and suppose $(x_1, x_2, \dots, x_n)$ is a permutation of $[n].$ Prove that the product of $(x_1-1)(x_2-2) \cdots (x_n-n)$ is even.

So far, I have this:
in order for the product to be even, we need to have an even number of odd integers $x_i$ and an odd number of even integers $x_j-j$.
But neither do I think this helps nor do I see a way of tying it up to arrive at a proof.

Furthermore, this section of the chapter involves the Pigeonhole Principle, so I’m sure the author wants us to incorporate that into each proof, but I can’t seem to do this either.

Any help would be much appreciated. 🙂 Thanks in advance.

Answer 1

Your pigeons are the odd $x_i$, your holes are the even $i$.