Binomial Theorem, Pascal & Combinatorial Identities · Superchallenge problem

A lattice path from $(0,0)$ to $(12,12)$ uses $24$ unit steps, each one step right or one step up. Every such path passes through exactly one lattice point $(a,\,12-a)$ on the line $x+y=12$. How many of these paths pass through such a point with $a$ even?