Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have a list of numbers, finite, about 50 and I want to know which permutations with subsets of that set sum up to a given number. I found a formula for the number of ways but I don't know how to find the actual numbers. Can you help me?

share|cite|improve this question
"which permutations with subsets of that set"? – Michael Zhao Oct 4 '12 at 19:05
Hope can help. – ziyuang Oct 4 '12 at 19:14
So, just to be clear, for example, you have {1, 2, 3, 4, 9, 10, 13, 18}, and you want to know how many ways you can add some subset together to get 23? And, you want to know the numbers involved as well, not just the number of ways to do it? – Graphth Oct 4 '12 at 19:15
It's similar to the subset-sum problem and might be transformed to that. As far as I can tell you have understood the question correctly. I think it is a nice question because it is easy to understand. – Programmer 400 Oct 4 '12 at 21:10
up vote 3 down vote accepted

I assume you mean you want to find the subsets of your list whose sum is a given number $S$. You can do it with dynamic programming. Suppose your numbers $x_1, \ldots, x_n$ are positive integers and the target sum is $s$. If $A_{k,m}$ for $1 \le k \le n$, $0 \le m \le s$, is the number of subsets of $\{x_1, \ldots, x_k\}$ whose sum is $m$, we have $A_{1,0} = A_{1,x_1} = 1$, $A_{1,m} = 0$ otherwise, and for $k \ge 2$, $A_{k,m} = A_{k-1,m} + A_{k-1,m-x_k}$ if $m \ge x_k$, $A_{k-1,m}$ if $m < x_k$.
Now if $A_{k,m} > 0$ we enumerate the subsets of $\{x_1,\ldots,x_k\}$ with sum $m$ recursively as follows: if $A_{k-1,m} > 0$, first enumerate the subsets of $\{x_1,\ldots,x_{k-1}\}$ with sum $m$; then if $m \ge x_k$ and $A_{k-1,m-x_k} > 0$ enumerate the subsets of $\{x_1,\ldots,x_{k-1}\}$ with sum $m-x_k$ and adjoin $x_k$ to each.

share|cite|improve this answer
Thank you for the answer. As far as I can tell you've understood the question correctly. – Programmer 400 Oct 4 '12 at 21:10

Your Answer


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.