An airport sees 1500 takeoffs per day. Prove that there are two planes that leave within a minute of each other.
All I can get started with is finding the total minutes in a day- 1440min. I understand that I have to fit 1500 planes into 1440 boxes however I do not know how to use this to prove that at least 2 planes leave within a minute of each other. Kindly help.
On
Imagine the day as divided into $1440$ one-minute intervals. Those will be your pigeonholes. The $1500$ planes are your (high-tech) pigeons. Think of a plane (= pigeon) leaving within a particular minute (= pigeonhole) as putting that pigeon in that hole. The pigeonhole principle says there must be two pigeons in a single hole. Given the nature of your pigeons, holes, and "pigeon being in a hole", this means that there must be two planes leaving in the same one-minute interval.
In general, when applying the pigeonhole principle to problems like this, you should look for ways to interpret "pigeon", "hole", and "pigeon being in a hole" in ways relevant to your problem, so that the conclusion of the pigeonhole principle, "at least two pigeons in the same hole" gets interpreted as the conclusion you want.
The Pigeonhole Principle seems to be the perfect approach.
Let the 1440 minutes be the pigeonholes and the 1500 takeoffs be the pigeons. It follows that $$\text{At least } \left\lceil\frac{1500}{1440}\right\rceil=2\text{ planes take off in the same minute}$$