# How many people would you need in a room to ensure with 100% probaility that three have the same birthday?

I am vaguely aware of the Pigeonhole principle and I understand that you would need 367 people to ensure that two people have the same birthday. I think that it may be required to have 734 people in a room to ensure 100% probability, since you can have 366 birthdays repeated and not have three people. Is this a correct assumption, and if so, how would you solve it without just guessing/checking?

• You mean 733.  – Did Apr 12 '15 at 16:08
• I think the use of "probability" in your question is misleading. There are situations in which "occurs with 100% probability" is weaker than "must occur", but this is not one of them. This is indeed a direct application of the Pigeonhole Principle. It sounds like you should become more than vaguely aware of the Pigeonhole Principle: certainly it is simple enough for anyone to understand, and once you get it you'll regard questions like this as being 100% (!) straightforward. – Pete L. Clark Apr 12 '15 at 17:16
• Any space large enough to contain that many people would probably not be considered to be a "room"; the word "hall" would seem more appropriate. – Marc van Leeuwen Apr 13 '15 at 10:35

There are $366$ possible birthdays; if we have $366\times 2=732$ people, then it is possible for each day, exactly two people call it their birthday. This is the maximum number of people where we can have at most $2$ per birthday. If we have one more person, then at least three people must share a birthday. Thus, in a room of $733$ people, at least three share a birthday.