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So I have this problem to answer and although I understand the concept, I have a little bit of difficulty solving this particularly worded one. Here it is:

Suppose you are walking in a rectangular grid from point $(0, 0)$ to the point $(m, n)$. In every step of the path, if you are at point $(x, y)$, you can either walk one step to the right, to $(x + 1, y)$ or a step up to $(x, y + 1)$. You are not allowed to move left or down.

a) What is the number of possible paths you can take?

b) If you choose a path at random, what is the probability that your path has exactly one turn?

I don't quite understand part b). For part a), I know that if a number was given, say $(3,2)$, then the total possible number of outcomes would result to ${5}\choose{3}$ or similarly ${5}\choose{2}$ since it would give you the same answer. Would it be fair to say that ${m+n}\choose{m}$ or ${m+n}\choose{n}$ would be a right answer?

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    Yes for part $a$. for $b$...well, how many paths have exactly one turn? – 2017-01-26
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    Well, you either turn right from all the way to the top, or go up from all the way right, and that would make 1 turn each. So 2 paths total? If that's right, then the answer should be ${5}\choose{2}/{m+n}\choose{m}$? – 2017-01-26
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    Right, so there are exactly $2$ such paths. Assuming they mean you choose each path with equal probability that makes $b's$ answer $\frac 2{\binom {m+n}m}$. – 2017-01-26
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    Ok got it! Thanks! :) – 2017-01-26

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