Count outcomes without listing them. Build from the multiplication principle to sampling with and without replacement and the binomial coefficient, then decide which count a problem needs. Finish with arrangements of repeated objects, the binomial theorem and sums of binomial coefficients.
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The multiplication principle
Learn how to count the outcomes of a multi-stage choice by multiplying the number of options at each stage, and when that applies.
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Sums of binomial coefficients
Learn how to use the binomial theorem to evaluate a sum of binomial coefficients and powers in a single step, by matching the sum to the expansion of a power of a sum.
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Lesson 01 · The multiplication principle
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The 9 lessons in this unit. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.
01
The multiplication principle
8 questions
Learn how to count the outcomes of a multi-stage choice by multiplying the number of options at each stage, and when that applies.
Covers
Counting sequential choices
When multiplication applies
02
Sampling with replacement
12 questions
Learn how to count outcomes when objects look alike by labelling them, count ordered samples drawn with replacement, and count the subsets of a set.
Covers
Labelling objects
Sampling with replacement
Counting subsets
03
Sampling without replacement
12 questions
Learn how to count ordered samples drawn without replacement, count the arrangements of a set of objects, and decide whether a scenario is sampled with or without replacement.
Covers
Sampling without replacement
Permutations
With or without?
04
The binomial coefficient
12 questions
Learn how to count selections where order does not matter by counting in order and correcting for the overcount, and how to name and evaluate that count as the binomial coefficient.
Covers
Adjusting for overcounting
Counting groups of
The binomial coefficient
05
Choosing a counting model
6 questions
Learn how to choose the right way to count a selection of objects by asking whether an object can be drawn more than once and whether the outcome records the order of the draws.
Covers
The sampling table
06
Properties of the binomial coefficient
12 questions
Learn how to write the binomial coefficient with factorials, use its symmetry, and find its value when choosing none, all or more than .
Covers
The factorial formula
Symmetry
Choosing none, all or more than
07
The multinomial coefficient
13 questions
Learn how to count arrangements of objects that include repeated copies, count the ways to split objects into labelled groups by choosing one group at a time, and name and evaluate that count as the multinomial coefficient.
Covers
Arrangements with repeated objects
Splitting into labelled groups
The multinomial coefficient
08
The binomial theorem
8 questions
Learn why the coefficients in the expansion of a power of a sum are counts of choices, and use the binomial theorem to write out the full expansion without multiplying the brackets.
Covers
Coefficients as counts
The binomial theorem
09
Sums of binomial coefficients
8 questions
Learn how to use the binomial theorem to evaluate a sum of binomial coefficients and powers in a single step, by matching the sum to the expansion of a power of a sum.
Covers
Sums with one power
Sums with two powers
This unit is part of our Essential Probability & Statistics for ML learning path, which contains 80 lessons. Every one of them is drawn below.
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9 lessons in this unit
12 prerequisite lessons across the unit, counting every step back to the start
59 other lessons in the learning path - after this unit, or alongside it
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