Question

In the 17th century, there were more than 800,000 inhabitants of Paris. At the time, it was believed that no one had more than 200,000 hairs on their head. Assuming these numbers are correct and that everyone has at least one hair on their head (that is, no one is completely bald), use the pigeonhole principle to show, as the French writer Pierre Nicole did, that there had to be two Parisians with the same number of hairs on their heads. Then use the generalized pigeonhole principle to show that there had to be at least five Parisians at that time with the same number of hairs on their heads.

Short answer

The resultant answer is that Pierre Nicole assumption was correct.

Step-by-step solution

Given data

There are more than 800,000 inhabitants of Paris is the given expression.

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

There were more than 800,000 inhabitants of Paris, and all of them had less than 200,000 hairs on their head, but no less than one hair. This means that there were more than 800,000 pigeons and less than 200,000 pigeonholes.

Thus, at least \(\left[ {\frac{{800,001}}{{199,999}}} \right] = 5\) people had the same number of hairs, which also proves that the French writer Pierre Nicole's assumption was correct.

Therefore, the Pierre Nicole assumption was correct.