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https://www.youtube.com/watch?v=Kl9WH5Eev2o Euler’s totient formula tells us the number of positive integers less than or equal to a given positive integer that are relatively prime to the latter. This is an important function in combinatorics and number theory. The function can be computed in a fairly simple fashion if we know the list of distinct prime […]

https://www.youtube.com/watch?v=fl2eoGBIT70 Given two finite sets, we can find the cardinality of their union by adding the two individual cardinalities and subtracting the cardinality of the intersection. But how can we find the cardinality of the union of $n$ finite sets? This is the general principle of inclusion-exclusion, otherwise known as PIE. We state this general

https://www.youtube.com/watch?v=2UHlMA-Qcg0 The multinomial theorem generalizies the binomial theorem by replacing the power of the sum of two variables with the power of the sum of an arbitrary finite number of variables $$(x_1+x_2+cdots+x_m)^n.$$ This is the result that gives rise to the concept of multinomial coefficients.

https://www.youtube.com/watch?v=5n8qXRTVUps In the binomial theorem, the goal is to expand the polynomial expression $(x+y)^n$ and find the coefficients after like terms have been collected. This is a very useful result in mathematics as a whole, and it is particularly useful in combinatorics for proving combinatorial identities.

https://www.youtube.com/watch?v=-yLJ8G_i_vM If we have $m$ cats and $n$ dogs, in how many ways can we choose $k$ of the animals? By iterating through the possible combinations of dogs and cats, we can derive a combinatorial identity, which is called Vandermonde’s identity. We derive this identity here using the described combinatorial or committee-forming proof.

https://www.youtube.com/watch?v=hk98kUxk-V0 The hockey stick identity in combinatorics tells us that if we take the sum of the entries of a diagonal in Pascal’s triangle, then the answer will be another entry in Pascal’s triangle that forms a hockey stick shape with the diagonal. Although proofs by induction or Pascal’s identity are possible, we show a

https://www.youtube.com/watch?v=jAcbcWV9CyE Similar to the question of summing a row of Pascal’s triangle, we can consider summing the squares of the entires of a row of Pascal’s triangle. By a combinatorial argument involving cats and dogs, we show that $$binom{n}{0}^2+binom{n}{1}^2+cdots+binom{n}{n}^2=binom{2n}{n}.$$

https://www.youtube.com/watch?v=1Ovnz74AasA If one takes the sum of a row of entries in Pascal’s triangle, one finds that $$binom{n}{0}+binom{n}{1}+cdots+binom{n}{n}=2^n.$$ In this video, we prove this remarkable combinatorial identity in two ways: by forming a committee and substituting numbers into the binomial theorem.

https://www.youtube.com/watch?v=fHZ7FniqPr4 If Pascal’s triangle is drawn out, one is immediately faced with the fact that the numbers in a row read the same from left to right as right to left. Since the entries of Pascal’s triangle are binomial coefficents, this observation is equivalent to the combinatorial identity $$binom{n}{k}=binom{n}{n-k}.$$

https://www.youtube.com/watch?v=Cwo2ua3H1pE Pascal’s triangle is a famous structure in combinatorics and mathematics as a whole. It can be interpreted as counting the number of paths on a grid, which is intimately linked with binomial coeffcients, otherwise known as combinations. This leads to a relationship between binomial coefficients, called Pascal’s identity, via a technique called double counting.