Let a1,...,a7 be an arbitrary arrangement of the numbers 1,...,7. Show that (a1-1)(a2-2)...(a7-7) is even.
Assume the contrary, i.e. is odd.
Then all of the factors etc. must be odd. Do you see why?
Since is odd, must be even.
Since is odd, must be odd.
Continue along these lines and see if you can arrive at an impossible situation, i.e. a contradiction.