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A little background: I thought about this problem yesterday, while I was helping a friend out at a particular competitive event where he was to cook a certain amount of food for guests, who would then vote for a winner. At this particular event, there were 450 guests being served by 15 food vendors (where my friend was 1 of the 15). The timespan of the event was 3 hours.

Now, the coordinators of the event told every vendor to make exactly 450 meals (it was a sandwich competition, so the meals were small sandwiches). This is obviously way to much; few people can eat 15 different meals in 3 hours. The more popular vendors will sell more than the less popular ones, however, for the sake of the problem, assume that each vendor is equally as popular as the next.

My question is this: 450 people attend an event. Each person eats at least one meal, with the maximum being 15 meals (no one is going to eat more than 15, because once they hit 15 they will have tried everything and thus know who to vote for). 15 vendors serve food to the 450 people at the event. How much should each vendor make in order to sell out (and thus not get stuck with a ton of leftovers)?

Extra info: My friend obviously did not sell out, and got stuck with a ton of leftovers. I think we sold around 150 sandwiches. Is the answer around 150-200? Just a rough guess. I really don't know how to solve it, or if there is even enough information to solve it, but was interested in the answer.

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1 Answer 1

If the sandwiches are small, a "normal" meal is probably a few of them. Would you guess 3-4? My experience is that when confronted with a large variety of free food, people eat less of each food, but probably 50% more in total than a normal meal. This comes from party settings, not competitions, so in your case it might be different. In this case, you would expect total sales of 675 meals, or (using a sandwich as 1/3 serving) 2000 sandwiches. This would say the average vendor sells about 133 sandwiches. Of course, some will serve more, some less. If I thought I was an average vendor, I would probably make 150-175, assuming the cost of having too many is less than the cost of having too few.

For a competition, it depends how it is structured. Professional judges will presumably just eat a bit of each sandwich, so you would need 450 to serve them all. Amateurs will probably keep eating the ones they like until they are full, then quit. It seems unlikely you have 450 pros, so I have assumed amateurs.

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