How many candies, of which there are $5$ types, must be distributed to a classroom to guarantee at least $40$ are distributed of the same type?

Question

Hi I have been searching how to do this question online but it feels like all the questions which are similar have an additional piece of information.

How many candies, of which there are $5$ types, must be distributed to a classroom to guarantee at least $40$ are distributed of the same type?

Any ideas or hints? Thank you!

Answer 1

I think this may be an example of what's called the pigeonhole principle. I like to think of it like balls and containers. Sometimes the abstraction for what should be considered the balls and what considered the containers isn't straight forward; if you think about the candy flavors here as the containers (or pigeonholes), how many people can you give a different candy to until you must repeat a flavor? You can only hand out five candies before you have to give someone a flavor that you've already given to someone else. You can extend this situation to your problem.

Answer 2

This is an application of the pigeonhole principle. If you want to avoid distributing $40$ any one type you can distribute $39$ of each. How many is that? Then add $1$.