Finding the number of shortest paths

Every path from A to B has to go through one of three points, call them P, Q and R, as shown below on the left.

Let's call a network of paths, like the one from A to P or from P to B, a "complete rectangle". If such a network consists of $m$ steps to the right, and $n$ steps upwards, the number of shortest paths would be $\dfrac{(m+n)!}{m!\ n!} = \dbinom{m + n}{m}$ . One way to see this is to label each step to the right as R, and every step upward as U; then each path is a permutation of $m$ R's and $n$ U's.

Then the number of paths from A to B that go through P is $\dbinom{6}{2} \dbinom{8}{3}$ ; the number of paths from A to B that go through Q is $\dbinom{6}{3} \dbinom{8}{4}$ ; and the number of paths from A to B that go through R is again $\dbinom{6}{2} \dbinom{8}{3}$ . Then the total number of paths is

$\binom{6}{2} \binom{8}{3} + \binom{6}{3} \binom{8}{4} + \binom{6}{2} \binom{8}{3} = \boxed{3080}$