Efficient algorithm to calculate the sum of all k-products

Question

Suppose you are given a list L of n numbers and an integer k. Is there an efficient way to calculate the sum of all products of

Answer 1

You can reduce k by 1:

e.g. for k=2

1*3 + 1*4 + 1*6 + 3*4 + 3*6 + 4*6

==

1*(3+4+6)+3*(4+6)+4*6

and for k=3

1*3*4 + 1*3*6 + 3*4*6

==

1*3*(4+6) + 3*4*6

So basically you cycle your list, then recurse to the same algorithm with k reduced by 1 and only the rest of the list