How many lattice paths are there from $(0, 0)$ to $(10, 10)$ that do not pass to the point $(5, 5)$ but do pass to $(3, 3)$?
What I have so far:
The number of lattice paths from $(0,0)$ to $(n,k)$ is equal to the binomial coefficient $\binom{n+k}n$ (according to Wikipedia). So the number of lattice paths from $(0, 0)$ to $(10, 10)$ is $\binom{20}{10}$, and the number of lattice paths through $(5, 5)$ is $\binom{10}{5}$, and the number of lattice paths that pass through $(3, 3)$ is $\binom{6}{3}$. What next?
Best Answer
Decompose your lattice paths in two parts: the one up until reaching $(3,3)$ and from $(3,3)$ to $(10, 10)$ not passing by $(5,5)$. We can translate this second part to be actually the number of paths from $(0, 0)$ to $(7,7)$ not passing by $(2,2)$. The fact that your paths must pass through $(3,3)$ make these problems independent.
For the first one we have ${6 \choose 3}$ possibilities
For the second one, the number of paths not passing through $(5,5)$ is all the paths minus those that do pass through $(5,5)$. We can calculate the second number using the same strategy splitting the path from $(3,3)$ to $(5,5)$ and then from $(5,5)$ to $(10, 10)$. This can be done in $${4 \choose 2}{10 \choose 5}$$ ways. So in total we get, for this second part $${14 \choose 7} - {4 \choose 2}{10 \choose 5}$$.
Since we can choose any path for the first part and combine it with any other one from the second, the nuumber we get is the product of the number for those parts which is $${6 \choose 3}\left({14 \choose 7} - {4 \choose 2}{10 \choose 5}\right)$$