Question 15
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 a $B$, then $w_{i+1}$ must be a $B$.
- $w$ 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 a $B$, then $w_{i+1}$ must be a $B$.