Let n be odd, let a be even, and let : {1, ..., n} + {1,..., n} be a permutation. Prove that the product (a +1 – 7(1))(a + 2 – 7(2)) ... (a +n – 7(n)) is even. Is the same conclusion necessarily true if n is even or if a is odd? Give a proof or a counterexamples in each case.