So we have totally N = m +n different objects. We have to select m things to form one group 1 and n things to form group 2.
1) Order of selection / order of formation of group is not important.
We take simply the number of combinations. Let us select m objects out of m+n objects for group 1. Automatically the group 2 is formed.
So: answer = C(m+n, m) or C(m+n, n) =
2) order of selection is important... So we take permutations of the members of the groups, in each group.
answer: C(m+n, m) * m! * n! = (m+n) !