If I have a $7 \times 5$ grid, and want to go from (0,0) to (7,5) we of course have $\binom{13}{7}$ possible lattice paths. lets say we want to find all lattice paths that pass through (1,4).
I thought this would simply by $\binom{5}{1}+\binom{7}{6}$ but this does not appear to be the correct solution
Furthermore if i wanted to find all paths that go through points (a,5-a) where a is an int from 1 to 5 (this should give me all possible paths) why is the following equation incorrect?
$$\sum_{i=0}^{5} {\binom{5}{i}+\binom{7}{7-i}}$$
Best Answer
The choices of paths from $(0,0)$ to $(1,4)$ and from $(1,4)$ to $(7,5)$ are independent. So multiplication principle applies.
For a given path from origin to $(1,4)$ you can take every possible path from $(1,4)$ to $(7,5)$. The options multiply. So correct are
$$\binom{5}{1} \cdot \binom{7}{6}$$ $$\sum_{i=0}^{5} {\binom{5}{i} \cdot \binom{7}{7-i}}$$