Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Question:

Consider a square matrix of order $m$. At each step you can move one step to the right or one step to the top. How many possibilities are to reach $(m,m)$ from $(0,0)$?

I think it is just counting the Central binomial coefficients.

Am I right? If not what is be the correct answer and why?

share|improve this question
1  
It is a square grid, rather than a matrix, most obviously because it has $(m+1) \times (m+1)$ points, though $m$ edges on each side. –  Henry Mar 8 '11 at 7:36
add comment

1 Answer 1

up vote 10 down vote accepted

You are correct. The reason is to get to $(m,m)$ you need to take a total of $m+m$ steps. However, you need to choose $m$ of those steps to be steps up, so the total number of paths is $\binom{m+m}{m}=\binom{2m}{m}$, since the central binomial coefficient picks which of the $2m$ steps will be up.

To add to this, in general, the number of paths from $(0,0)$ to $(m,n)$ is then $\binom{m+n}{m}=\binom{m+n}{n}$ for the same reasoning.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.