Shared Birthday

Question

Using the Pigeonhole Principle, determine the minimum number of people required to ensure that at least one of the twelve months has 7 or more people born in it. Specifically, how many people are needed to guarantee that there is at least one month with at least 7 people born in it?

Accepted Answer

To find the minimum number of people required to guarantee that at least one month has at least 7 people born in it, we use the Pigeonhole Principle.

There are 12 months in a year. To avoid having any month with 7 or more birthdays, the maximum number of people per month would be 6. If we have 6 people per month, we would have:
$6 * 12 = 72$ people.

With 72 people, each month could have exactly 6 people born in it, avoiding the situation where any month has 7 or more birthdays. To ensure that at least one month has 7 people born in it, we need one more person:
$72 + 1 = 73$ people.

Therefore, the minimum number of people required to guarantee that at least one month has 7 people born in it is: 73.

Title	Category	Subcategory	Difficulty	Status
Catching Ants	Brain Teasers	Pigeonhole Principle	Easy	Unlock
Handshakes	Brain Teasers	Pigeonhole Principle	Medium	Unlock
Pair of Socks	Brain Teasers	Pigeonhole Principle	Easy	Unlock

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