You will always move a total of m-1 RIGHT and n-1 DOWN. The order is not important. So lets say you choose your path by pulling from a hat pieces of paper with either DOWN or RIGHT printed on them. There are m-1 RIGHT’s and n-1 DOWN’s. Combinatorics tells us that there are (m+n-2)! different orderings for this selection, but it is made up of two types in the amounts m-1 and n-1. Thus, the count is:

(m+n-2)!
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(m-1)! (n-1)!