# A variation of how many integer solutions are there for $a+b+...=n$

There was a combinatorics question last night that I thought I could answer, turns out I couldn't (so I discarded my draft) but I thought I could, I now want to find out if I got close.

The question was about splitting 50 things into any number of groups of 1,2,3,4 or 5 items.

If I were asked how many integer solutions there were to $A+B+C+D+E=10$ say I'd be quite happy to conclude that you "encode" any such solution using... say X and -s, so "XXX-XXX--XXX-X" denotes 3As, 3Bs, 0Cs 3Ds and 1 E, any arrangement of 10Xs and 4 dashes is a solution. So it is simply $\frac{14!}{10!4!}$ using the axiom of choice and such, I am happy with this work.

You can also modify it to work out how many integer solutions there are when any number is greater than a certain value (by substituting something like $A=a+1$ then A>=0 means a>=1).

But I cannot use this to solve A+2B+3C+4D+5E=n, at least not directly, I had a play on paper but I could not find anywhere to go with this, was I on the right lines or is this a totally different problem?

• The Axiom of Choice??? Feb 17, 2014 at 14:02
• @MarcvanLeeuwen Sorry I should have stated, given a decision with m outcomes and another independent with n the number of ways to choose is nm. So in this case you have a string of 14 things, there are 14 options for the first, 13 for the second ... 14! in total, of those 10! will be duplicates with respect to X (as you can't actually tell the Xs apart) and 4! duplicates with the dashes. Think of it as each string of 14! contains a substring of 10 Xs and 4-s, there will be 10! forms of that very substring and such. Feb 17, 2014 at 14:05

You are asking for the number of partitions of the number $50$ into parts of size at most $5$. It is equal to the number of partitions of the number $50$ into at most $5$ parts, and is the coefficient of $X^{50}$ in the power series $\prod_{i=1}^5\frac1{1-X^i}$, but there is no easy formula for such coefficients.

There is a lot of literature about counting partitions though.

• Could you give me a little more info about where to find out more, and what this type of function is called? I like how X's value plays no part, that's interesting. I'd like to know where I can find out more. Feb 17, 2014 at 15:37
• There is already quite a bit at Partition in Wikipedia. Feb 17, 2014 at 17:05

$A+2B+3C+4D+5E=n$ is the same problem, and it says to divide n in groups of multiple of 1's, multiple of 2's ...upto multiple's of 5's

Which is given by coefficient of $x^n$ in the expansion of

$$(1+x+x^2+\cdots)\cdot(1+x^2+x^4+\cdots)\cdot(1+x^3+x^6+\cdots)\cdot(1+x^4+x^8+\cdots)\cdot(1+x^5+x^{10}+\cdots)$$ $$={(1-x)^{-1}}\cdot(1-x^2)^{-1}\cdot(1-x^3)^{-1}\cdot(1-x^4)^{-1}\cdot(1-x^5)^{-1}$$

• This is twice you've given some allegedly correct answer with no explanation or source as to what magic you have done to get it!! Feb 17, 2014 at 15:38
• @AlecTeal I think it's self-explanatory. Nice Jun 15, 2018 at 17:16