Question

Please help to understand the concept of
The no. Of ways in which (m+n) different things can be divided into two groups which contain m and n things respective
Please also mention its case also
1) if order of group is not important
2)if order of group is important

Answer 1

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) = ${}^{m+n}C_m={}^{m+n}C_n$

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) !
       ${}^{m+n}C_m\ *\ m! *\ n! =\frac{(m+n)!}{m!\ n!}*m!*n!=(m+n)!$