Question

My problem is, I need to count how many combination of array of integers sums to a value W.`

let say:

int array[] = {1,2,3,4,5};

My Algorithm is just find all combinations of lengths from 1 to W / minimum(array), which is equal to W because minimum is 1.
And checking each combination if its sum equal to W then increment a counter N.

any other algorithm to solve this ? should be faster 🙂

Update:
ok, the subset problem and the Knapsack Problem are good, but my problem is that the combinations of the array repeats the elements, like this:

1,1,1 -> the 1st combination
1,1,2
1,1,3
1,1,4
1,1,5
1,2,2 -> this combination is 1,2,2, not 1,2,1 because we already have 1,1,2. 
1,2,3
1,2,4
1,2,5
1,3,3 -> this combination is 1,3,3, not 1,3,1 because we already have 1,1,3. 
1,3,4
.
.
1,5,5
2,2,2 -> this combination is 2,2,2, not 2,1,1 because we already have 1,1,2. 
2,2,3
2,2,4
2,2,5
2,3,3 -> this combination is 2,3,3, not 2,3,1 because we already have 1,2,3.
.
.
5,5,5 -> Last combination

these are all combinations of {1,2,3,4,5} of length 3. the subset-sum problem gives another kind of combinations that I’m not interested in.

so the combination that sums to W, lets say W = 7,

2,5
1,1,5
1,3,3
2,2,3
1,1,2,3
1,2,2,2
1,1,1,1,3
1,1,1,2,2
1,1,1,1,1,2
1,1,1,1,1,1,1

Update:
The Real Problem is in the repeated of the elements 1,1,1 is need and the order of the generated combination are not important, so 1,2,1 is the same as 1,1,2 and 2,1,1 .

Answer 1

No efficient algorithm exist as of now, and possibly never will (NP-complete problem).

This is (a variation of) the subset-sum problem.