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 $\left(\dfrac{a_1+1}{2}\right)\left(\dfrac{a_2+2}{2}\right)\cdots \left(\dfrac{a_6+6}{2}\right) > 6!$