A Permutation Teaser

Algebra Level 5

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

How many permutations satisfy that \[\left(\dfrac{a_1+1}{2}\right)\left(\dfrac{a_2+2}{2}\right)\cdots \left(\dfrac{a_6+6}{2}\right) > 6!\]