Listing and counting outcomes

Foundation and Higher

The idea

List outcomes in a fixed order so none is missed or repeated. When there are $m$ choices for one thing and $n$ for another, there are $m \times n$ combinations.

Fix the first choice and run through every second choice.
Move to the next first choice and repeat.
Check the count: $3$ starters and $4$ mains give $3 \times 4 = 12$ meals.

Watch out Decide whether order matters: AB and BA are one pair of people but two different codes.

See it worked

Worked exampleNon-calculator3 marks

A café sells three kinds of sandwich: cheese (C), ham (H) and egg (E). It sells two drinks: tea (T) and juice (J). Lucy buys one sandwich and one drink.

(a) List all the possible combinations Lucy could buy.

(b) How many different combinations are there?

(a) Work through the sandwiches in order, pairing each with every drink: CT, CJ, then HT, HJ, then ET, EJ.
(b) Count the list: $3$ sandwiches each with $2$ drinks makes $6$ .

Answer (a) CT, CJ, HT, HJ, ET, EJ; (b) $6$

Your turn

Question 1Non-calculator4 marks

Ella has four coins in her pocket: 10p, 20p, 50p and £1. She takes out two coins at random.

(a) Work out how many different pairs of coins she could take out.

(b) How many of the pairs have a total value of more than 75p?

Question 2Non-calculator4 marks

Use two different digits from 1, 2, 4, 6, 8 to form a two-digit number.

(a) How many of the numbers are even?

(b) How many of the numbers are odd?

Question 3Non-calculator5 marks

Three positive whole numbers add up to $12$ . The order does not matter, so $2 + 3 + 7$ and $7 + 2 + 3$ count as the same set.

(a) Work out how many different sets of three numbers there are.

(b) Three sticks have these whole-number lengths, in cm, and are used as the sides of a triangle. For how many of the sets from part (a) can a triangle be made?

In past papers

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