What is Combinatorics? When we hear about combinatorics or combinatorial problems, there is a lot of talk about balls and boxes and other objects of the physical world. But where does all that fit into the mathematical universe? In essence, combinatorics is finite set theory, which has two tasks: proving that a set is finite, and computing …

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Counting the Pages of a Book How many pages of a book are there between p. 10 and p. 40, inclusive? This is a problem where it is easy to make an off-by-one error, as the answer is not $$40-10=30.$$ We show how to solve it, along with generalizations to arithmetic and geometric sequences.