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

An interesting property you could explore is the distributive property of multiplication in relation to addition.

L=[a,b,c,d]

When k = 1, it's trivial:

S=a+b+c+d

When k = 2:

S = a * (b + c + d) + b * (c + d) + c * d

When k = 3, things get a bit more interesting:

S = a * b * ( c + d) + (c * d) * (a + b)
S = a * (b * (c + d)) + c * d) + b * (c * d)  <-- this form lends itself better to the algorithm

And for k = 4:

S = a * b * c * d

This should hold for larger values of n.

An implementation in C#:

 private static int ComputeSum(int[] array, int offset, int K)
 {
     int S = 0;

     if (K == 1)
     {                      
         for (int i = offset; i < array.Length; i++)
             S += array[i];
     }
     else if ((array.Length - offset) == K)
     {
         S = array[offset] * ComputeSum(array, offset + 1, K - 1);
     }
     else if (offset < array.Length)
     {
         S = ComputeSum(array, offset + 1, K) + array[offset] * ComputeSum(array, offset + 1, K - 1);             
     }

     return S;
 }

Which can be further improved by memoization.