how many persons need to be in one room , so AT LEAST 3 people have the same Birth MONTH?

Question

how many persons need to be in one room , so AT LEAST 3 people have the same Birth MONTH ?

Another Answer i suggest : (correct me if i'm wrong) there should be 15 People at least , because if there's more people than months per year , then using Pigeonhole Principle at least two people are born in the same month.

What have you tried? Usually one approaches these problems by trying to think of the largest group without the desired property. — lulu, Apr 03 '19 at 18:36
Maybe related:https://math.stackexchange.com/questions/25876/probability-of-3-people-in-a-room-of-30-having-the-same-birthday and https://www.pleacher.com/mp/mlessons/stat/bday2.html — NoChance, Apr 03 '19 at 18:39
Your reasoning is correct, and it is formalized in the Pigeonhole Principle. — Austin Mohr, Apr 03 '19 at 18:41

score 1 · Answer 1 · answered Jun 24 '25 at 00:26

Just so there is an answer

If the question is "how many people must there be to guarantee there is a month in which at least three people all have birthdays" then the answer is $25$ people. With $2\times 12=24$ people or fewer you could have up to $2$ people having birthdays in each of the $12$ months, but with $25$ people this becomes impossible by the pidgeonhole principle.
If the question is "how many people must there be to guarantee there are at least three people who share a birthday month with someone else" then the answer is $14$ people. With $13$ people you could have $2$ people sharing a birthday month and $11$ not sharing; with $14$ there must either be one month with at least $3$ people sharing that month, or two months each with at least two people sharing that month in which case at least $4$ people sharing a birthday month with someone else.

You tagged this with birthday which is usually about probability.

If the question is "how many people must there be to have a probability of at least $\frac12$ that there is a month in which at least three people all have birthdays" then the answer is $11$ people ($10$ people gives a probability of $\frac{477503}{995328}\approx 0.48$ which is smaller than $\frac12$).
If the question is "how many people must there be to have a probability of at least $\frac12$ that there are at least three people who share a birthday month with someone else" then the answer is $8$ people ($7$ people gives a probability of $\frac{383}{768}\approx 0.4987$ which is slightly smaller than $\frac12$).

how many persons need to be in one room , so AT LEAST 3 people have the same Birth MONTH?

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