how to solve this question using two loops?

[James Tursa]

Once you have the formula you can calculate this with MATLAB functions. Other than that, what is the MATLAB related question here? I assume this is homework, so we would appreciate you showing some effort first.

[John D'Errico]

Anyway, why do you need MATLAB to do this at all? Seems like pencil and paper are sufficient to write down the general formula. Oh, yeah, its homework. So start writing code. Or, think about the problem, and then write down the solution as a simple enough formula.

I'd suggest first thinking along the lines of symmetry. Is there some symmetry in this problem? What does that mean in terms of the solution? (Often symmetries in mathematics are useful tools to reduce the complexity of a problem.)

As well, look at small problems, then see if there is a pattern you might recognize.

Answer 1

[James Tursa]

Big Hint:

Suppose the grid is M x N. You've got (M+N) total moves to make, with M of them being "down" and N of them being "right". Suppose you number the moves 1,2,...,(M+N). How many ways can you pick M numbers from this ordered list of (M+N) numbers for the ordering of the "down" moves (the remaining numbers will be the "right" moves)? Figure that out, and what MATLAB function can be used to calculate it, and you will have your answer.

[John D'Errico]

Interestingly, my solution is completely different, arising from a different approach, but with a simple identity can be shown to be identical.

Answer 2

[Image Analyst]

Steve Eddins gives an extensive discussion on this in his 5-blog series on this: http://blogs.mathworks.com/steve/2011/11/01/exploring-shortest-paths-part-1/