Compute the number of possible words $w = w_1w_2\cdots w_{100}$ satisfying:

• $w$ has exactly $50$ $A$’s and $50$ $B$’s (and no other letters).
• For $i = 1,\ 2,\ \cdots,\ 100$, the number of $A$’s among $w_1$, $w_2$, $\cdots$, $w_i$ is at most the number of $B$’s among $w_1$, $w_2$, $\cdots$ , $w_i$.
• For all $i = 44,\ 45,\ \cdots ,\ 57$, if $w_i$ is an $B$, then $w_{i+1}$ must be an $B$