GCSE · Maths · Edexcel · Spec 1MA1 · Higher

Systematic listing strategies

How many 3-letter passwords can you make from 26 letters? Make a guess now. By the end, you'll count them without writing a single one down.

Maths · Listing outcomes

Every meal on the menu

Pick a starter, then a main, then a dessert, and see which meal you land on. Then go back and walk the paths in order, top to bottom.

Starter → Main → Dessert

  • Choosing a whole meal is one trial with three stages. Each finished meal is one outcome. The set of every possible meal is the sample space.

2 × 3 × 2 = 12 possible meals, and the tree ends 12 times.

On Choose a meal. 2 branches to choose from.

Key: S soup, G garlic bread, P pasta, C curry, B burger, I ice cream, F fruit. Read the ends from top to bottom and you have the complete list: ξ = {SPI, SPF, SCI, SCF, SBI, SBF, GPI, GPF, GCI, GCF, GBI, GBF}.

Exam line: You can count without drawing: multiply the number of options at each stage. Here 2 × 3 × 2 = 12. A menu with 3 starters, 4 mains and 2 desserts gives 3 × 4 × 2 = 24 meals.
Watch out: Writing meals down as they come to mind (SPI, GCF, SBF…) is how one gets missed or written twice. Fix the first choice, run through everything after it, then move on.

Same outcomes, two pictures

Two-way tablevsTree diagram

Example: pick one letter from A, B, C and one number from 1, 2, 3, 4.

Focus

How you build it

Two-way table

Letters A, B, C down the side, numbers 1 to 4 across the top.

Tree diagram

A branch for each letter, then a branch for each number from every letter.

The insight

Both make you pair every first option with every second option. That is what makes them systematic.

What one outcome looks like

Two-way table

One cell: row B meets column 3 to give B3.

Tree diagram

One full path: B, then 3, to give B3.

Reading off the total

Two-way table

Count the cells: 3 rows × 4 columns = 12.

Tree diagram

Count the ends: 3 branches, each splitting into 4, gives 12.

Adding a third choice

Two-way table

Rows and columns are used up. You would need a separate table for each option of the third choice.

Tree diagram

Grow another set of branches from every end.

Check the list

The total says 20. Is the list right?

Maya takes one letter card from A, B, C, D and one number card from 1, 2, 3, 4, 5. She lists every possible outcome. Which line goes wrong?

Maya's list — which line goes wrong?

Exam line: After listing, check the total AND how often each option appears.

Guess first

How many 3-letter passwords?

A password is 3 characters long. Each character can be any of the 26 letters, and letters can repeat. How many different passwords are possible?

Your estimate

15000 passwords

0 passwords30000 passwords

Read the context

Which calculation fits?

For each situation, ask: can the same item be chosen twice? Does it matter which one is first? Then pick the calculation.

Still to sort

n × n (0)

Repeats allowed: every stage still has all n options.

Where the line is: Differs from n × (n − 1) only in whether the first item can be chosen again.

n × (n − 1) (0)

No repeats, and first and second are different roles.

Where the line is: Differs from the ÷ 2 case only in whether swapping the two gives a new outcome.

n × (n − 1) ÷ 2 (0)

No repeats, and the two just make a pair.

Where the line is: n × (n − 1) counts AB and BA separately. For a pair they are the same, so halve it.

6 of 6 still to sort.

Watch out: The numbers alone never tell you. Ask the two questions every time: can it repeat, and does order matter?

Your turn

Two stages, one answer

A pizza deal lets you choose 2 different toppings from 5 and 2 different dips from 10. The order you pick them in doesn't matter. How many different deals are there?

  1. Toppings: 5 choices for the first, 4 left for the second, so 5 × 4 = 20.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Fix the first choice, run through every option after it, and nothing gets missed or counted twice.

What you need to know

  • A trial is a single test that may have one stage or several. An outcome is one result of it. The sample space is the set of all possible outcomes, written like ξ = {H, T}.
  • List systematically: fix each option of the first choice in turn and pair it with every option of the next choice (A1, A2, A3, … then B1, B2, …).
  • A systematic list, a two-way table and a tree diagram show the same set of outcomes.
  • Product rule: to count the outcomes when one item is chosen from each group, multiply the numbers of options in each group.
  • Read the context first: no repeats means the next stage has one fewer option; if the order of a pair doesn't matter, divide by 2.

The big picture

When a choice has several stages, list the outcomes in a fixed order: hold the first choice still, pair it with every option of the next choice, then move on. A list, a two-way table and a tree diagram all show the same set of outcomes. Check a list by its total and by how often each option appears. To count without listing, multiply the number of options at each stage, after first asking whether items can repeat and whether order matters.

Key points

1Fixing the first choice and running through every later choice is what stops outcomes being missed or repeated.
2To add a stage to a list, copy the whole list once for every option of the new stage: a 3-sector spinner gives 3, then 9, then 27 outcomes.
3Abbreviations such as S for soup are fine as long as you give a key.
4Check a list twice: the total, and how often each option appears. With 4 letters and 5 numbers, each letter appears 5 times and each number 4 times.
5The same number of options at every stage gives a power: 3 coin flips give 2³ = 8 outcomes, and 4 rolls of a six-sided dice give 6⁴ = 1296.
6Stages counted separately combine by multiplying: 10 ways for one stage and 45 for another give 10 × 45 = 450.

Worked example

Problem

The digits 1, 2, 3 and 4 are written on four cards. Two cards are placed side by side to make a 2-digit number. List every possible number systematically, then check your total.

⚠ Watch out

Treating every count as 'just multiply the options'. Picking 1st and 2nd from 5 people is 5 × 4 = 20, not 5 × 5, because nobody can take both places. If they only form a pair, it's 5 × 4 ÷ 2 = 10.

🧠

Memory hook

Hold one, run the rest, move on. Then multiply to check the total.

✓

Check yourself

List every outcome of choosing X, Y or Z and red or blue, in fixed-first-choice order. Does it have 3 × 2 = 6 entries, with each letter twice and each colour three times?

Flashcards

(14)
What is a trial?
A single test decided in advance. It can have one stage (one spin of a spinner) or several (flipping a coin three times).
What is an outcome?
One possible result of a trial, such as one complete meal from a menu.
What is the sample space, and how is it written?
The set of all possible outcomes of a trial, listed inside curly brackets, e.g. ξ = {H, T} for one coin flip.
How do you list outcomes systematically?
Fix each option of the first choice in turn and pair it with every option of the next choice: A1, A2, A3, … then B1, B2, …
Name three ways to display all the outcomes.
A systematic list, a two-way table or a tree diagram. All three show the same set of outcomes.
In a two-way table of outcomes, what does one cell show, and how do you get the total?
One cell is one combination (its row option with its column option). Total = rows × columns.
How do you extend a list to one more stage?
Copy the whole list once for every option of the new stage and pair each copy with that option. A 3-sector spinner: 3, then 9, then 27.
When can you use letters like S and P for outcomes?
Any time, as long as you give a key saying what each abbreviation means.
Two ways to check a finished list?
The total is right, AND each option appears the expected number of times (4 letters × 5 numbers: each letter 5 times, each number 4 times).
What is the product rule for counting?
When one item is chosen from each of several groups, multiply the numbers of options: 3 starters, 4 mains, 2 desserts give 3 × 4 × 2 = 24.
How many outcomes for 4 rolls of a six-sided dice?
The same 6 options at each of 4 stages gives a power: 6⁴ = 1296.
Choosing two items without replacement: what changes?
The second stage has one fewer option: 6 × 5 = 30 for two draws from 6 cards, compared with 6 × 6 = 36 with replacement.
When do you divide by 2?
When two items are chosen and their order doesn't matter, because the product counts each pair twice: 52 × 51 ÷ 2 = 1326 ways to take two cards at once.
How do you combine stages you counted separately?
Multiply the stage counts: 10 ways for one stage and 45 ways for another give 10 × 45 = 450 ways to do both.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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