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Counting the Power Set of a Finite Set

https://www.youtube.com/watch?v=MW7_7OAe330 If we know that the cardinality of a finite set is $n$, what is the cardinality of its power set? It turns out that there is a simple formula, which is $2^n$. We prove this fact in this video using the weak multiplication principle and bijection principle from combinatorics.

Casework and Complementary Counting

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 …

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What is Combinatorics?

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 …

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Vector Cross Product Derivation

https://www.youtube.com/watch?v=gpO4eW0kP_k Students in their senior high school year often learn the formula for the cross product of two three-dimensional vectors by heart. In this video, we derive the cross product formula from first principles and show an amazing way of remembering it using the determinant of a formal matrix.

Trigonometric Dot Product

https://www.youtube.com/watch?v=8iWPQKzqYII The dot product is a natural function that can be defined on a pair of Euclidean vectors. Interestingly, it can be combined with the cosine law to provide insight into the angle (or that angle’s explementary angle) between two vectors. Thus, we can derive what can be called the trigonometric form of the dot …

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Euclidean Triangle Inequality

https://www.youtube.com/watch?v=aZLZWJZ4cdQ In math, there are various triangle inequalities, ranging from the classical one that actually applies to triangles to one for the complex numbers. In this video, we derive the triangle inequality related to the Euclidean norm and inner product for tuples of real numbers.

Regular Octahedron: Surface Area & Volume

https://www.youtube.com/watch?v=8O8zwFx6mJI After the regular tetrahedron, the regular octahedron is the next simplest of the Platonic solids. In fact, the octahedron can be constructed by gluing together two square-based pyramids on their square faces. In this video, we present and prove simple formulas for calculating the surface area and volume of a regular octahedron in terms …

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