I want to show that:
$$\prod_{j}(1+a_{j})-\prod_{j}(1+b_{j})=\sum_{j}(\prod_{i < j}(1+a_{i})(a_{j}-b_{j})\prod_{k > j}(1+b_{k}))$$
The only hint I have is that I can replace $$ a_j - b_j = (1-a_j ) + ( 1-b_j) $$
I see there are $a_j$ and $-b_j$ terms after expansion of the RHS of the equation but I do not understand how the cross terms become zero.