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. $\displaystyle (a_1 - 1)(a_2 - 2) \cdots (a_7 -7)$ is odd.
Then all of the factors $\displaystyle (a_1 - 1), (a_2 - 2)$ etc. must be odd. Do you see why?
Since $\displaystyle a_1 -1$ is odd, $\displaystyle a_1$ must be even.
Since $\displaystyle a_2 -2$ is odd, $\displaystyle a_2$ must be odd.
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Continue along these lines and see if you can arrive at an impossible situation, i.e. a contradiction.