Counting sounds easy. You start at one and keep going.
But what if we want to count possibilities?
Suppose you have:
- 3 T-shirts: red, blue and green
- 2 pairs of trousers: black and grey
How many different outfits can you make?
Counting systematically
We could simply start listing outfits:
So there are 6 different outfits.
With larger problems, simply writing down possibilities as they occur to us can be dangerous.
Can we be sure that we counted every possibility exactly once?
A system for counting
Suppose we make a two-letter code using only the letters A, B and C. Letters are allowed to repeat.
How many different codes are possible?
Instead of writing codes down randomly, organise them according to their first letter.
AA
AB
AC
BA
BB
BC
CA
CB
CC
There are 9 codes.
Notice that AB and BA are different. Here, order matters.
When counting possibilities, look for a system that makes it clear that nothing has been missed and nothing has been counted twice.
An ice-cream shop has three flavours: chocolate, strawberry and vanilla.
You can have your ice cream in either a cone or a cup.
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Reveal answer
There are 6 possibilities:
- chocolate cone
- chocolate cup
- strawberry cone
- strawberry cup
- vanilla cone
- vanilla cup
Tables
Sometimes a table makes the structure even easier to see.
Imagine four sandwich fillings and two kinds of bread.
| White | Brown | |
|---|---|---|
| Cheese | Cheese + white | Cheese + brown |
| Egg | Egg + white | Egg + brown |
| Hummus | Hummus + white | Hummus + brown |
| Peanut butter | Peanut butter + white | Peanut butter + brown |
The table contains 8 boxes, so there are 8 possible sandwiches.
Tree diagrams
Another useful way of organising choices is a tree diagram.
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Each complete route through the tree represents one possible outfit.
Notice how the diagram makes all the possibilities visible without relying on us to remember which ones we have already counted.
A safe uses a three-digit code.
Each digit can be 1, 2 or 3, and digits may be repeated.
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121 122 123
131 132 133
Reveal answer
There are 9 codes beginning with each possible first digit.
So altogether:
9 + 9 + 9 = 27
There are 27 different codes.
Does order matter?
Suppose Alice and Ben are sitting in two numbered seats on a bench.
These are different arrangements, because Alice and Ben are sitting in different seats.
But suppose instead that Alice and Ben are simply chosen to form a team. The team {Alice, Ben} is exactly the same team as {Ben, Alice}.
Does changing the order create a new possibility?
Alice, Ben, Charlie and Dom each shake hands with every other person exactly once.
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Reveal answer
Alice has 3 handshakes:
AB, AC, AD
Ben has 2 new ones:
BC, BD
Charlie has 1 new one:
CD
3 + 2 + 1 = 6
There are 6 handshakes.
Alice–Ben and Ben–Alice are the same handshake, so counting both would count the same thing twice.
Five points are drawn on a page. No three lie on the same straight line.
A triangle can be made by choosing any three of the points.
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Reveal answer
The triangles containing A are:
ABC, ABD, ABE, ACD, ACE, ADE
That’s 6.
The triangles without A are:
BCD, BCE, BDE, CDE
That’s another 4.
6 + 4 = 10
So there are 10 different triangles.
Later in this course we’ll learn a much quicker way to answer questions like this.
What have we learned?
Counting is not always about arithmetic. Often the hard part is organising the possibilities.
Three useful tools:
- systematic lists;
- tables;
- tree diagrams.
Two dangers:
- missing a possibility;
- counting the same possibility twice.
And before you count, always ask: does order matter?
Next: Choices in stages
Listing possibilities works when there aren’t too many of them. But what if there are hundreds, thousands or millions?
In the next lesson we’ll discover a simple idea that lets us count them without listing them at all.
What did you think?
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