Pigionhole Principle

Question

Among any group of 3000 people there are at least 9 who have the same birthday. I cant figure out what's the object is and what's the box. And, how to apply it in the principle

Answer 1

Take $366$ possible birthdays according to the Christian calender:

$31+29+31+30+31+30+31+31+30+31+30+31$.

Divide $3000$ people into $366$ groups, so that the maximum number of people who share the same birthday is minimal:

• Put $\Bigl\lfloor\frac{3000}{366}\Bigl\rfloor=8$ people per group
• Take the remaining $3000-366\cdot8=72$ people, and put each one in a different group

The result is:

• $ 72$ groups of $9$ people per group
• $294$ groups of $8$ people per group

Hence there are at least $9$ people who share the same birthday.

Answer 2

You put the people (objects) into days (boxes) of the year. The way to get the least people with the same birthday, is by distributing the people evenly over the days. This means that we calculate $3000/366\approx8.20$. That means there will be a day (box) with at least 9 people (objects) in it.

Answer 3

Let the boxes be the dates of birth (that's 366 if we count leap years, and 365 if not), and let the pigeons be the people. From there, it's simple calculations