Marvin the fly starts at $(0,0).$ Each step, Marvin moves one unit right or one unit up. He is trying to get to the point $(5,7)$. However, at $(4,3)$ there is a frog that will eat him if he goes through that point. In how many ways can Marvin reach $(5,7)$?
+874 Success = Total - Not Success
\(\begin{align*} \binom{5+7}{5} - \binom{4+3}{4}\binom{1+4}{1} &= \binom{12}{5} - \binom{7}{4}\binom{5}{1} \\ &= 792 - (35 \cdot 5) \\ &= 792 - 175 \\ &= \boxed{617} \end{align*}\)
thank you, i subtracted by $35$ because i did not multiply by 5 choose 1, mind if i ask why you multiplied by that?
@MathProblemSolver101 i think ur wrong...why should you subtract 7 choose 4 times 5 choose 1? I understand you if the question says that you cant go to (4, 3), but the question says there is a frog there. He will get eaten if he goes there so no more journey!!! Maybe check again?
P.S. CPhill can you check?
You can also use waterfall/waterflow if you learn/already know about it.
