Method wanted

Consider an $m\times n$ rectangular grid . Find the total number of paths one can reach from lower left corner to upper right corner .

Plz post ur method and other variations possible in such questions,

For example the number of shortest path possible from one corner to opposite corner is $\frac{(m+n)!}{m!\times n!}$

I dont know how to solve if its asked number of paths possible to reach from one corner to the corner above it.

#Combinatorics

Easy Math Editor

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Comments

Assume person starts from lower left corner. To take the shortest path, one can travel only up or right in each step. And there are of course, $m+n$ steps to take. In the end, the person is at the top right corner, this means that he/she has traveled $m$ units up and $n$ units right. The order in which these steps were arranged is the thing that matters here and is the thing we have to count. Basically you need the coefficient of $x^ny^m$ in $(x+y)^{m+n}$ .