 # Q. Prove that the number of ways in which (m + n) things can be divided into two groups containing m and n objects respectively is (m + n)!/(m! n!).

10 years ago

As we separate m+n items into m items and n items separately we need to know that in how many ways we could separate m+n items into distinct sets of m and n things.So to do that let''s find out first in how many ways we could take m things from (m+n) things.The total no of ways in which we could do that would be (m+n)Pm (i.e) (m+n)!/n!.Now the rest of the items automatically go to the set of n things.

But the m things in the first separated set repeat themselves many times by just changing positions in the set of m things.Each of the things could occupy anyone of the m places in the first set.The total ways they repeat is m!.Therefore to eliminate repetitions in the sets of m and n things we divide (m+n)!/n! by m!.

Simply put it''s just (m+n)Cm.Since there is no repetition of things (by changing positions in each set) in m set it becomes unique and so does n set as the rest of the things go into it.Hence the result.