A collection of coins contains six different denominations: pennies, nickels, dimes, quarters, half-dollars, and dollars. How many coins must the collection contain to guarantee that at least 100 of the coins are of the same denomination?
To me, there can be infinitely many coins in the collection and there is still no guarantee that at least 100 of the coins are of the same denomination. What presumption am I missing here? This is the way the question appears in a chapter on the Pigeonhole principle.
If there are infinitely many coins, then there are at least $100$ available of some type. If they're all allocated to a single denomination, for instance, that type has at least $100$ coins. If the infinity of them is distributed amongst multiple types, then the same is true. If I have infinitely many apples, then I do have $10$, and $100$, and $1,000$, and $10^{100}$ of them that I could give away to anyone.
Think of it like this to solve it: what is the worst-case scenario you can envision that still meets the constraints of not having $100$ or more coins of the same type? Obviously, just less than that, $99$ - for every type. From there, any additional coin allocated to any particular type will constitute $100$ and thus ensure at least one type has $100$.