# How many up-or-to-the-right paths from $(0,0)$ to $(8,5)$, and related questions

Recall that in the section on functions, we created an encoding for the paths from the point $(0, 0)$ to the point $(m, n)$ which either went right or up at each step. In particular, we said that each path can be represented as a bit string of length $m+ n$ with exactly $m$ 0's and $n$ 1's.

For this problem, let $m = 8, n = 5$.

a. How many paths are there from $(0, 0)$ to $(8, 5)$?

b. How many of these paths go through the point $(3, 3)$?

c. How many of them avoid the point $(3, 3)$?

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Unless I miss my guess, you have copied out a problem from some source without attribution, which constitutes plagiarism. Please let us know where this problem comes from; why it's worth working on; what progress you have made on it. If it's homework, please add the "homework" tag. –  Gerry Myerson Jul 29 '12 at 23:56
–  Gerry Myerson Jul 29 '12 at 23:59
I got this question from one of my university professor website. the univ is UWM. I have a Qualifying exam this September and I am solving as much as I can from Discrete mathematics. the source is : –  Sam Jul 30 '12 at 0:02
cs.uwm.edu/classes/cs317 –  Sam Jul 30 '12 at 0:02
These things are called Dyck paths. –  Raphael Jul 30 '12 at 8:43
a. How many ways are there of arranging eight $0$s and five $1$s in a string?
b. How many ways are there of arranging three $0$s and three $1$s in a string? How many ways are there of arranging five $0$s and two $1$s in a string? How many ways are there of arranging three $0$s and three $1$s followed by five $0$s and two $1$s?