HMMT 2017 C10

Compute the number of possible words $w=w_{1} w_{2} \ldots w_{100}$ satisfying:

• $w$ has exactly $50 A$ 's and $50 B$ 's (and no other letters).
• For $i=1,2, \ldots, 100$, the number of $A$ 's among $w_{1}, w_{2}, \ldots, w_{i}$ is at most the number of $B$ 's among $w_{1}, w_{2}, \ldots, w_{i}$.
• For all $i=44,45, \ldots, 57$, if $w_{i}$ is an $B$, then $w_{i+1}$ must be an $B$.