The product of $(a_1-1)(a_2-2)....(a_{169}- 169)$ is

Question

Let $a_1,a_2,.....a_{169} $ represent any arbitrary permutation of the number $1,2,3....169$. Then the product $(a_1-1)(a_2-2)......(a_{169} - 169)$ is

1. Odd only for some permutation, not all
2. Always even, whatever be the permutation
3. Always odd, whatever be the permutation
4. Even only for some permutation, not all

I have no idea how to solve this problem .

Answer 1

There are more odd than even indices, hence some $a_{\text{odd}}$ must be odd, thus making the product even.