Let's assume set $$S = (2,3,4,5)$$ I like to express the sum of the products of all possible unique combinations of $k$ (distinct) elements of $S$. I know the binomial coefficient is equal to:
$${\displaystyle {n \choose k}={\frac {n!}{k!(n-k)!}} }$$
I want to represent the following summations in a more general mathematical formula:
$$X_1 = (2) + (3) + (4) + (5) $$ $$X_2 = (2 . 3) + (2 . 4) + (2 . 5) + (3 . 4) + (3 . 5) + (4 . 5)$$ $$X_3 = (2 . 3 . 4) + (2 . 3 . 5) + (2 . 4 . 5) + (3 . 4 . 5)$$ $$X_4 = (2 . 3 . 4 . 5)$$
My best try is:
$$X_k = \sum_{i}^{k} S_i . \dots . S_k$$
It doesn't feel right... Anyone who can help me?