# How to write the sum of all possible combinations of $k$ variables chosen from $n$ values using sigma notation?

I am trying to write a general expression for the equivalent resistance of $$n$$ parallel resistors. Of course the well-known formula is

$$R_{eq} = \left(\dfrac{1}{R_1} + \dfrac{1}{R_2} + \cdots + \dfrac{1}{R_n}\right)^{-1}.$$ In the case of two parallel resistors, this can be simplified to a product-over-sum expression:

$$R_{eq} = \dfrac{R_1R_2}{R_1+R_2}.$$

When more than 2 resistors are in parallel, it's easier to just use the first equation, but for fun I decided to try to derive a product-over-sum expression for $$n$$ resistors. This is quite simple to derive but I'm having trouble with how I would express the sum in the denominator. In English, the denominator turns out to be "the sum of all possible combinations of $$n - 1$$ resistances"; for example, if we have 4 parallel resistors, their equivalent resistance is

$$R_{eq} = \dfrac{R_1R_2R_3R_4}{R_1R_2R_3 + R_1R_2R_4 + R_1R_3R_4 + R_2R_3R_4}.$$

The numerator is easy to express using product notation, but like I said I don't know how to express the denominator in the general case of $$n$$ resistors. I expect it would look something like this:

$$R_{eq} = \dfrac{\prod\limits_{i=1}^{n}R_i}{\sum\limits_{i=1}^n(\text{something...})}.$$

Also it would be interesting to generalize the denominator to not just take all possible combinations of $$n-1$$ quantities, but all possible combinations of any number $$k$$ of quantities.

Quite trivial if you have seen some conditions/inequalities below the $$\sum$$ or the $$\prod$$ symbols
The expression in the denominator you are looking for is $$\sum_{i=1}^n\prod_{\substack{j=1 \\ j\neq i}}^n R_j$$
For the generalization to the 'sum taken $$k$$ at a time', let us find the sum with respect to the set $$S={x_1,x_2,...,x_n}$$
We write $$\sum_{\substack{{T\subseteq S} ,\\ {|T|=k}}}\prod_{x\in T} x$$