Combinatorics

https://www.youtube.com/watch?v=53zDF3lv0tw The standard pigeonhole principle tells us that stuffing too many pigeons into too few holes will result in a hole that has more than one pigeon. But we can make this statement far stronger and more precise. In fact, in this video, we derive the strongest possible pigeonhole results so that you can apply […]

https://www.youtube.com/watch?v=RUYPgm8JFMs Casework or combinatorial addition allows us to split a set into disjoint pieces or bring together disjoint pieces into a larger set. Combinatorial subtraction or complementary counting, tells us that we can find the cardinality of a larger set and subtract the excess. Both are indispensable techniques, especially when combined with more sophisticated methods

https://www.youtube.com/watch?v=3ZPpcIzn3bQ 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

https://www.youtube.com/watch?v=DZ8XOS8qJDU 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.