Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

What are the number of ways of dividing $n_1$ objects of type $1$, $n_2$ objects of type $2,\ldots,n_k$ objects of type $k$ into $2$ parts?

share|improve this question

2 Answers 2

up vote 2 down vote accepted

I think it is:$$\prod_{i=1}^{k}(n_i+1)$$ The reasoning is that for each i you can keep either ${0,1,2,....n_i}$ balls in the first part and the rest of the balls in the 2nd part so there are $(n_i+1)$ choices for each i. From this the result follows. This also includes the case when one part is empty.

share|improve this answer
    
I think we should subtract one from the product to exclude the case when each i is assigned "0" right?? –  user60576 Feb 3 '13 at 16:52

It's solved by first finding the $N$ "Permutations of Like Objects," and then applying the binomial coefficient.

Definition: The total number of permutations of all $n$ distinct objects for which $r_1$ are alike, $r_2$ are alike $\ldots, r_k$ are alike is

\begin{equation} N=\frac{n!}{r_1!r_2!... r_k!} \end{equation}

Then use the binomial coefficient to find the number of ways of selecting 2 objects from your $N$ permutations using

\begin{equation} N_{total}=\binom N 2= \frac{N!}{2!(N-2)!} \end{equation}

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.