## Discrete Math: Assumption: The world's population is seven billion people and everyone is 120 years or younger. State any further assumptions you make in the following:

a. What is the maximum number of people in the world that must share a birthday?

b. Prove that there are at least 111 people whoe were born at exactly the same minute on the same date of the same year.

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We use the Extended Pigeonhole Principle. We assume there are 365 days in a year, 24 hours in a day, and 60 minutes in an hour, therefore 525,600 minutes in a year. Since the maximum age is 120 years, we therefore have 525,600*120=63,072,000 minutes one could possible be born. We therefore have n=7,000,000,000 people allocated to m=63,072,000 minutes

a. By Pigeonhole Principle, there must be at least one birthday with no more than floor(7,000,000,000 / (365 * 120)) = 159,817 people sharing the same birthday.

b. By Pigeonhole Principle, there are at least ceil(7,000,000,000/63,072,000) = 111 people born during the same minute.

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