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

Question

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

Answer 1

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

Answer 2

$1$ 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 $$\prod _{i=1}^k (a_{i1}+a_{i2})$$

Addendum:

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

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