Three couples sit for a photograph in 2 rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many arrangements are possible?

There are $$6! = 720$$ seatings in all, and the forbidden events are the three couples occupying a horizontally adjacent pair or a vertical pair. Counting the seatings that avoid all of these, by inclusion and exclusion over the three couples, leaves 96. Hence 96 arrangements are possible.