# Pigeon-hole principle help

#### vertex

##### New member
My teacher gave my class a practice sheet that we can use to study for our exam and on it has this problem.

A Community College hosts an alumni reunion and exactly 300 people attend the reunion. Each guest in attendance is asked to enter their birth date on a laptop during the sign-in registration. The computer tracks not only the year and month that each person was born, but also determines the day of the week that their birthday fell on. As it turns out, the guests can be partitioned so that there are exactly the same number born in each of the decades of the1950’s, 60’s, 70’s, 80’s, and 90’s.

a.) Must there be at least one day of the week for which at least 43people were born on that day (at least 43 born on a Monday, or at least 43 born on a Thursday, etc.)? Explain your reasoning using the PIGEON-HOLE PRINCIPLE

I'm not really sure were to start. Ive tried looking at notes but i'm not that good at solving word problems.

#### Dr.Peterson

##### Elite Member
Suppose that for each day of the week, there were less than 43 people born on that day. How many people could there be, total?

I don't think this requires the pigeonhole principle; I imagine subsequent questions will.

#### vertex

##### New member
Why is it less than 43? It says at least 43, so shouldn't that mean greater than or equal to 43?

#### vertex

##### New member
After reading the pigeon-hole principle, I think its supposed to be like this.

The 300 people born represent the pigeons. The days of the week that the people are born on represent the number of pigeon holes. The number of people is larger than the number of days so at least 43 people must point to the same day. This means that at least 43 people are born on the same day.

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#### pka

##### Elite Member
A Community College hosts an alumni reunion and exactly 300 people attend the reunion. Each guest in attendance is asked to enter their birth date on a laptop during the sign-in registration. The computer tracks not only the year and month that each person was born, but also determines the day of the week that their birthday fell on. As it turns out, the guests can be partitioned so that there are exactly the same number born in each of the decades of the1950’s, 60’s, 70’s, 80’s, and 90’s.
a.) Must there be at least one day of the week for which at least 43people were born on that day (at least 43 born on a Monday, or at least 43 born on a Thursday, etc.)? Explain your reasoning using the PIGEON-HOLE PRINCIPLE
Hint: $$42\times 7=~?$$