Or even "approximately" 200, roughly 200, near 200, anything that tells you it's pretty darn close to 200 exactly.

Yeah, but it sounds weird. It make it seems like they wanted to fire 200 due to budget restraints, but seeing as they already need to do the paper work, might as well fire that one guy who just annoying. Can't very well come out and say: We're firing 200 people from various departments, oh and Travis. He's gone as well.

It raise all sorts of weird questions. Was it three departments that was just shutdown, and that happens to be 201 people? Did they have a target salary cut, which just fit nicely with these exact people?

In reality it will always be some odd number of people getting laid off. It's just that when numbers get to specific it sounds like there's a point. Try telling someone that you want to lose 23,5kg. People will ask why 23,5?. Say you want to lose around 20kg, much fewer questions.

what if they lay off another person, does the article need to be update then? There's some advantage in using approximate numbers.

I'd have phrased it as "two hundo"