Determining number of lattice paths Question:
Determine the number of lattice paths from $(0,0)$ to $(6,6)$ that take steps in $ \{\ (1,0) , (0,1) \}\ $ that do not go through the point $(3,3)$.
I'm not sure what my professor means by the "that take steps in" because these steps imply that he walks backwards. What does he mean?
I know how to find the amount of lattice paths from one end to another by simply doing $x+y $ choose $x \text{ or } y$. But what do I do when a point is excluded?
 A: Hint: The easiest way is to find the number of paths from $(0,0)$ to $(6,6)$, then subtract those that go through $(3,3)$.  You say you can do the first, then find the number of ways to get from $(0,0)$ to $(3,3)$.  For each one of those, how many ways are there to get from $(3,3)$ to $(6,6)?$
A: First of all, I believe the problem is trying to say a valid move is to increase either the first or second coordinate by $1$.  In other words, from $(3,2)$, you could go to $(4,2)$ or $(3,3)$.
As for how to solve it, can you figure out how many paths do go through $(3,3)$ and exclude them?
A: The length of the word is (6,6) = 6+6 = 12. Calculate all the possibility from (0,0) to (6,6) with the formula or (12!/6!(12-6)! = 924.
Because you may not go through (3,3) then calculate from (0,0) to (3,3). (Now is the length of the word 6 because (3,3) = 3+3 or 6. so (6!/3!(6-3)! = 20. But the length from (3,3) to (6,6) is equal from (0,0) to (3,3) so you get 20*20 or 400. Then you have the total possibilities that are 924 - 400 = 524 possibilities from (0,0) - (6,6) without go through (3,3). That is your answer: 524
