Binomial Coefficient

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This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.

This is #Day13 in the series.

#AdventofMathematicalSymbols
#Analysis
#Calculus
#Mathematics

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

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Fantastic explanation, clearing up my own misunderstanding the day before a final exam in university. Dear thanks to you!

TheDreamRemains

For those who are interested: the binomial coefficient can be generalized so that the upper index in the symbol is actually any complex number, not just an integer. In fact, it can be generalized to be anything you want it to be, as long as it is part of a ring. How does it work? Well, again, notice that the binomial coefficient is equal to n!/[k!·(n – k)!]. Let us just focus on the symbol n!/(n – k)! for a second. This is called the falling factorial of order k, of n. What is nice about the falling factorial is that it can be expressed as a polynomial with respect to n: this polynomial has degree k, and is called the kth Stirling polynomial. Now, why is this important? It is important because polynomials can have complex inputs, matrix inputs, or whatever input you want them to have, as long as you are not changing the degree, and as long as the thing you are inputting belongs to a ring (so that addition and multiplication are well-defined and associative and addition is commutative). So if I substitute n for a complex number z in the kth Stirling polynomial, and I divide by k!, then that results in the binomial coefficient z choose k. This is extremely useful, as it helps express a generalization of the binomial theorem.

angelmendez-rivera

Awesome. Very well explained in just under 4 minutes. Thank you very much

hannes

Great explanation! Thanks, will help me with a math proofs course

RealEverythingComputers

A very important symbol! Thanks for all your videos! 😃

punditgi

Best video i could fint on this subject, thanks man :D

gormenorm

This explained a concept my biostats professor has had me confused on for 3 weeks….in 3 minutes…. Omg

hanaleabray

The way you can think about the final equation is for each arrangement of the numbers you did choose, you're also going to have every arrangement of the numbers of you didn't choose, so the denominator becomes k!(n-k)!.

This actually presents an interesting insight: n choose k is the same as n choose (n - k), because multiplication is commutative.

ryan-tabar

Hi, there is a little mistake in your notation of the binomial coefficient at 2:46, the multiplication series above goes on from (n) till and including (n-k-1), not till (n-k+1)

suleymenkand