A Permutation Teaser

Algebra Level 5

Let {a1,a2a6}\{a_1,a_2\ldots a_6\} be a permutation of {1,2,6}\{1,2,\ldots 6\}.

How many permutations satisfy that (a1+12)(a2+22)(a6+62)>6!\left(\dfrac{a_1+1}{2}\right)\left(\dfrac{a_2+2}{2}\right)\cdots \left(\dfrac{a_6+6}{2}\right) > 6!

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