Serge Lang's Basic Mathematics: Binomial Coefficiants

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(m, n) + (m, n-1) = (m +1, n)

I have read as many explanations of this problem that I could find and I still have no clue how to do it. I understand that m! can be written as m(m-1)!, and likewise for the rest of the problem but I don't know how to get the common denominator that is listed in the answer for this problem in the back of the book: n!(m−n+n)! . I am tearing my hair out over this and would appreciate any guidance. Thanks.