# Thread: This is a basic combinatorics problem

1. ## This is a basic combinatorics problem

Dear All,

I will highly appricitate your help. The problem is attached as a PDF, pleas take a look.

Regards

2. ## Re: This is a basic combinatorics problem

[QUOTE=Rafael;714765]The problem is attached as a PDF, pleas take a look./QUOTE]
First a comment in the form of a question. Why do you expect us to open another file? You could easily type the question yourself.
The question is: You have N bottles and n balls. You need to calculate the number of all possible distributions (configurations, states) of balls in the bottles considering that all balls and bottles are identical.
Now here is another critical comment: the formula in that post has nothing to do with the solution of the problem as it is stated. Because the bottles are identical as are the balls this is simply a question of how one partitions the integer n into N or fewer summands.
Here is an example: $\displaystyle n=4~\&~N=3$.
We could have: $\displaystyle n=4~,3+1,~2+2,~2+1+1,~\&~1+1+1+1$ that is four ways.

Now there is no neat closed solution to this question. All solutions to the integer partition problem are recursively defined sequences (functions).
Here are some examples: if $\displaystyle n=10~\&~N=10$ then $\displaystyle P(10,10)=42$, while $\displaystyle n=10~\&~N=4$ then $\displaystyle P(10,4)=23$.

Now I actually question the use of the word identical in the original question. The suggested answer implies otherwise.

3. ## Re: This is a basic combinatorics problem

Dear Plato thanks for your reply. I am new here and I decided to attach the problem since I already
had it as a PDF in my computer.
According to you one can to find the number of states analytically
as a function of N and n.
Identical in the original question means that for example balls can not be distinguished from one another.
May I ask you for reference of your solution if there is any, in order to understand it in more detailed way.

4. ## Re: This is a basic combinatorics problem

Originally Posted by Rafael
May I ask you for reference of your solution if there is any, in order to understand it in more detailed way.
Look at MATHEMATICS OF CHOICE By Ivan Niven chapter six.

5. ## Re: This is a basic combinatorics problem

Thanks for suggesting the book.
[QUOTE/Now there is no neat closed solution to this question. All solutions to the integer partition problem are recursively defined sequences (functions)./QUOTE]
So the recursive expression about which you were talking is PN(n)=PN-1(n)+PN(n-N) for 1<N<n right ?
Now regarding point b. in my problem "you have two different types of balls let us say n blue and m red balls, and you need to partition them into N identical bottles ".
What can you suggest for this case? In chapter 8.2 of above mentioned book there is being discussed problem of distribution of m different elements into k identical boxes.
G(m,k) denoted as the number of distributions of m different things into k identical boxes. And g(m,k) denoted the number of distributions of m different objects in k identical boxes, with no empty boxes. is it right to say that g(m,k) = G(m-k,k) (assuming that m>k>1) ?
Thanks in advance.