Question

Show that if there are 100,000,000 wage earners in the United States who earn less than 1,000,000 dollars (but at least a penny), then there are two who earned exactly the same amount of money, to the penny, last year.

Short answer

The resultant answer is at least 2 people who get the same wage.

Step-by-step solution

Given data

The given expression is 100,000,000 wage earners in the United States who earn less than 1,000,000 dollars.

Concept of Pigeonhole principle

If the number of pigeons exceeds the number of pigeonholes, at least one hole will hold at least two pigeons, according to the pigeon hole principle.

Simplify the expression

Every dollar is 100 pennies, so \(1,000,000\)dollars \( = 100,000,000\) pennies. There are \(100,000,000\)people who earn less than \(100,000,000\) pennies. This means that there are \(99,999,999\)possible wages (pigeonholes) and 100,000,000 people (pigeons).

By the Pigeonhole Principle, there are at least \(\left[ {\frac{{100,000,000}}{{99,999,999}}} \right] = 2\) people who get the same wage.

Therefore, at least 2 people who get the same wage.