On a symmetrical matrix how do you calculate the sum of every possible combination of 3 elements ?

I have a 10X10 Symmetrical Matrix, Where the Columns and Rows are both S1 to S10 shown below,

What I need is a code that will get me the sum of every possible combination of 3, example: S3,S4,S8 = sum of S3S4 ,S3S8, S4S8

[John D'Errico]

Can you have repeated indices? I.e., {2,2,5}?

[Avishka Karunaratne]

No you can't

Answer 1

[Walter Roberson]

M = dec2bin(0:2^10-1,10) - '0';
mask = sum(M,2) == 3;
M = M(mask,:);
sums = M * S;

[Walter Roberson]

By the way, my estimate is that this is cost-effective up to roughly 22 elements in the original matrix, after which the memory requirements slow it too much. There are other ways to write it that would use less memory.