Module 1
Counting successive choices
Prerequisites: Multiplication of natural numbers
What you will learn: Build and count all outcomes of successive choices using tree diagrams and the fundamental counting principle.
Learning paths · Combinatorics
A laboratory-first path from the fundamental counting principle to factorials, permutations, arrangements and combinations, with and without repetition, ending with autonomous model selection and a bridge to probability.
It is ready and unlocks 2 more skills in the learning path.
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Prerequisites: Multiplication of natural numbers
What you will learn: Build and count all outcomes of successive choices using tree diagrams and the fundamental counting principle.
Prerequisites: Fundamental counting principle
What you will learn: Interpret n! as a descending product and use it to count complete orderings.
Prerequisites: Factorial
What you will learn: Recognize problems where all distinct elements are used and order matters, calculating P_n=n!.
Prerequisites: Simple permutations and factorial
What you will learn: Remove duplicates caused by repeated elements using n!/(n1!n2!…).
Prerequisites: Simple permutations
What you will learn: Count ordered selections of k distinct elements chosen from n using D(n,k)=n!/(n-k)!.
Prerequisites: Fundamental counting principle
What you will learn: Count length-k sequences when each position may reuse any of n symbols: n^k.
Prerequisites: Simple arrangements and factorial
What you will learn: Count groups of k elements chosen from n when order does not matter using the binomial coefficient.
Prerequisites: Simple combinations
What you will learn: Count unordered selections with repetition using stars and bars and C(n+k-1,k).
Prerequisites: Permutations, arrangements and combinations
What you will learn: Independently decide whether order, repetition and use of all elements lead to permutations, arrangements or combinations.
Prerequisites: Choosing a combinatorial model
What you will learn: Combine counting ideas and use favourable/possible outcomes to prepare probability calculations.
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