The number 1 rule in in catering or feeding people for that matter is don't run out of food. So it's good that you're expecting 180 ppl but are planning for 200 people. I would guess that a regular guy would eat 2 of the 3 meat choices you mentioned. So to make it simple. 2 portions per person X 200 ppl=400 portions. 400 portion /3 meats=133.33 portions of each meat. You can round up or down as you see fit. That may be alot of food if you have sides. You also have to think will it be a buffet or will you dish out the food. People tend to take more if they are serving themselves. I have found doing catering that people are more than happy to take leftovers if they're avaiable. Hope this helps. Let is know how it goes.
18.5" WSM, 22.5" Weber Kettle, Weber Smokey Joe