I want to write in a correct and intelligible mathematical formalisation that I'm doing the sum of all the possible combinations of the product of $k$ probabilities within a set of $n$ probabilities (that sum to one).
Example: Let $S$ be a set of $n=4$ probabilities, e.g., $S={0.2,0.3,0.4,0.1}$, the unique possible combinations of $k=3$ elements where at least one element differs from the others are $\{0.2,0.3,0.4\}$,$\{0.2,0.2,0.3\}$, $\{0.2,0.3,0.1\}$, $\{0.2,0.4,0.1\}$, ... etc.
How would you write it, knowing that the set of probabilities may vary in values and in number of elements?
Difficulties for me not having a degree in mathematics are: 1/ how to write a combinations of $k$ elements of a set of $n$ elements such as in the $k$ elements at least one differs from the other? 2/ I'm not sure -even pretty sure- I can use the word "set" and "subset" about "a set of probabilities". Thank you for help indicating correct vocabulary and way to formalize such quite simple algorithm idea.
Here are two ways to write such a sum of products: $$\sum_{A \subset S: |A|=k} \prod_{i\in A} p_i$$ $$\sum_{\substack{A \subset S:\\ |A|=k}} \prod_{i\in A} p_i$$