Overview
On this page, we continue our study of Combinatorics by discussing combinations, Pascal’s Triangle and the Binomial Theorem.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
- Understand the definition of \({n \choose k}\) in terms of factorials and how it counts unordered selections.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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Be able to count the number of subsets of a fixed size and solve related counting problems using combinations.
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Understand the connection between counting binary strings, the binomial theorem, the value of \({n \choose k}\), and Pascal’s triangle.
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Be able to prove identities involving binomial coefficients algebraically and with combinatorial arguments.
To prepare for class
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Watch the following video (by Kolumath) which explains combinations and how to count them:
After class
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Watch the following video (by Wajdi Mohamed Ratemi) which showcases Pascal’s Triangle and some of its well-known (and not so well-known) applications, including the Binomial Theorem:
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Read the explanations on this page for some more details about Pascal’s Triangle, the Binomial Coefficients \({n\choose k}\), and the Binomial Theorem.
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Watch the following video (by NancyPi) which shows how to use the Binomial Theorem in practice:
