# Notation for all combinations of choosing one element for each of $k$ sets

I have $$k$$ pairs of elements, and I want to take a summation over a product of the $$2^k$$ different ways to choose one item from set. Can someone suggest a good notation for this?

So for example, if I had 3 sets of two elements $$\{a_{11},a_{12}\}, \{a_{21},a_{22}\}, \{a_{31}, a_{32}\}$$ I would want

$$a_{11}a_{21}a_{31} + a_{11}a_{21}a_{32} + a_{11}a_{22}a_{31} + a_{11}a_{22}a_{32} + a_{12}a_{21}a_{31} + a_{12}a_{21}a_{32} + a_{12}a_{22}a_{31} + a_{12}a_{22}a_{32}$$

Edit: while several people have pointed out that it is equal to $$\prod_{i=1}^k(a_{i1}+a_{i2})$$ I need to work on the expanded form. I was more looking for a way to notate the individual $$2^k$$ $$k$$-tuples that are all the combinations of $$\{1,2\}^k$$

Thanks, Craig

• $\prod\limits_{i=1}^k(a_{i1}+a_{i2})$ Commented Mar 28, 2020 at 18:15

$$1$$ element is chosen from each of the $$k$$ sets. Lets say there are $$3$$ sets. Then the sum is $$(a_{11}+a_{12})(a_{21}+a_{22})(a_{31}+a_{32})$$.

Then for $$k$$ such sets, the sum will be $$\prod _{i=1}^k (a_{i1}+a_{i2})$$

A more generalized expression with $$n$$ sets and $$m$$ elements in each set can be written similarly

$$\prod_{i=1}^n \sum_{j=1}^m a_{ij}$$

$$\sum_{j\in \{1,2\}^k}\;\prod_{i=1}^ka_{ij_i}$$