PigeonholePrinciple

Problem - 2128
Let $a_1, a_2, a_3, \cdots a_n$ be a randomly ordered sequence of 1, 2, 3, . . . , $n$. Prove the following product is an even number if $n$ is an odd integer: $$(a_1 -1)(a_2-2)(a_3-3)\cdots(a_n-n)$$

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