James the fly starts at (0, 0) Each step, James 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 James reach (5, 7)?

Guest May 15, 2021

#1**+1 **

For obvious reasons, $(0,0)\to (m,n)$ can be done in $\binom{m+n}{m}=\binom{m+n}{n}$ ways. We let $f(m,n):=(0,0)\to (m,n)$.

So we just want $f(5,7)-f(4,3)f(5-4, 7-3)$, which is trivial to compute using the expression for $f(m,n)$.

thedudemanguyperson May 15, 2021

#3**+1 **

James the fly starts at (0, 0) Each step, James 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 James reach (5, 7)?

First I am going to consider how many ways assuming there is __no__ fly eating frog

Each step must be up or right. There are 5 steps to the left and 7 steps up.

so there are 12C5 or 12C7 ways = 792ways

Now how many of these will be unfinished because the frog will get its supper?

that would be 7C3* 5C1 = 35*5 = 105 paths for the frog to get its supper.

So there are 792 - 105 = 687 safe routes.

Melody May 15, 2021