Find all possible row-wise sums in a 2D array

Question

Ideally I'm looking for a c# solution, but any help on the algorithm will do.

I have a 2-dimension array (x,y). The max columns (max x) varies between 2 and 10 but can be determined before the array is actually populated. Max rows (y) is fixed at 5, but each column can have a varying number of values, something like:

   1 2 3 4 5 6 7...10

A  1 1 7   9 1 1
B  2 2 5   2 2
C  3         3
D            4
E            5

I need to come up with the total of all possible row-wise sums for the purpose of looking for a specific total. That is, a row-wise total could be the cells A1 + B2 + A3 + B5 + D6 + A7 (any combination of one value from each column).

This process will be repeated several hundred times with different cell values each time, so I'm looking for a somewhat elegant solution (better than what I've been able to come with). Thanks for your help.