# Thread: a tough combinatorics problem

1. Find the number of different combinations of $\displaystyle r$ natural.numbers that add upto $\displaystyle n$

I tried this for quite a fair amount.of.time but.couldn't figure it out.

2. Originally Posted by mathworker
Find the number of different combinations of $\displaystyle r$ natural.numbers that add upto $\displaystyle n$

I tried this for quite a fair amount.of.time but.couldn't figure it out.
This is known as the problem of Partitions of an Integer.
.

If you want print material see Ivan Niven's Mathematics of Choice, chapter six.

Thanks for the link,I have gone through it.
But as far as I understood partition function doesn't give the number of partitions of specific cardinality.I mean if we want only the partitions that contains $\displaystyle r$ terms for example or can we define a restricted partition function that can do the job?If we can define how can we approximate such restricted $\displaystyle p(x)$

4. Originally Posted by mathworker
Thanks for the link,I have gone through it.
But as far as I understood partition function doesn't give the number of partitions of specific cardinality.I mean if we want only the partitions that contains $\displaystyle r$ terms for example or can we define a restricted partition function that can do the job?If we can define how can we approximate such restricted $\displaystyle p(x)$
Well I did say that the webpage is only fair. I dislike its notation.
I suggest that you try to find Niven's book.

Example: The number of partitions of 6 into 3 summands is three:
\displaystyle \begin{align*} 6 &= 1+1+4\\ &=1+2+3\\ &=2+2+2\end{align*}

That is $\displaystyle p_3(6)-p_2(6).$

There is a clear recursive definition of $\displaystyle p_k(n).$

5. Originally Posted by mathworker
Find the number of different combinations of $\displaystyle r$ natural.numbers that add upto $\displaystyle n$
There is about this. Of course, the problem is tricky if "combination" is used in its technical sense to mean a set. In contrast, permutations (i.e., ordered sequences) of summands are called compositions (rather than partitions). Their number is .

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