How can I simply write the expression :$y_1 y_2 x_3 x_4x_5+ y_1y_3x_2x_4 x_5+\cdots$

Question

How can I simply write the expression: $$y_1 y_2 x_3 x_4x_5+ y_1y_3x_2x_4 x_5+\cdots=$$ using $\sum $ and $\prod$? Any help will be appreciated. Thanks.

Answer 1

Whether this really counts as "simple" is perhaps open to debate: $$ \sum_{S\subset\{1,2,3,4,5\}\atop |S|=2}\;\,\prod_{i\in S}y_i\prod_{i\in\{1,2,3,4,5\}\setminus S}x_i$$