Walking Down The Tethered Grid

Could u pls explain ur diagram. I solved using principle of inclusion and exclusion.

We can use inclusion-exclusion. Let's label the lower-left cross P, the upper right as S, and the others as Q and R.

The number of paths from a point M to N, where N is x over and y up from M, is (y+x)!/(y!*x!). We can see this because we must arrange x "overs" and y "ups" in a permutation to define a path, so you are permuting y+x steps with x repeated symbols and also y repeated symbols.

There are 252 paths from A to B.

There are 2 paths from A to P, and 70 from P to B, meaning 140 go through A. There are 5 ways paths A to Q, and 5 from Q to B, meaning 25 go through Q. Symmetrically, 25 go through R and 140 through S.

There are 2 paths from A to P, 1 from P to Q, and 5 from Q to B, meaning 10 go through P and Q. Symmetrically, 10 go through P and R. There are 2 paths from A to P, 20 from P to S, and 2 from S to B, meaning 80 go through P and S. There are 5 paths from A to Q, 1 from Q to S, and 2 from S to B, meaning 10 go through Q and S. Symmetrically, 10 go through R and S.

There are 2 paths from A to P, one from P to Q, 1 from Q to S, and 2 from S to B, meaning 4 go through P, Q and S. Symmetrically, 4 go through P, R and S.

Using inclusion exclusion, we count: 252 - 140 - 140 - 25 - 25 + 10 + 10 + 80 + 10 + 10 - 4 - 4 = 34