Systematic Listing: Edexcel GCSE Foundation maths revision notes

What systematic listing means

Systematic listing means writing down every possible combination or outcome in a fixed order, so that none is missed and none is written twice.

Suppose you choose one drink, tea or juice, and one snack, a biscuit or a banana. In a random order it is easy to miss one. In order, you get all four: tea and biscuit, tea and banana, juice and biscuit, juice and banana.

In the exam you are usually asked to list all the possible combinations, such as one drink and one cake, or all the outcomes when a coin is thrown three times, for 2 marks. Other questions ask for every code made from a letter and a number, every order three people could finish a game in, a table of the scores from two spinners, or how many numbers you can make with some cards. For full marks you need every combination, with none repeated and no extras.

Fix one choice at a time

Use the letters the question gives you. If J stands for juice and M for muffin, write that combination as JM. Keep each combination clearly separate: put a comma between groups, or put each group in brackets, such as (J, M).

Keep the first choice fixed and go through every second choice in order. Then move on to the next first choice, and go through the second choices in the same order again. Carry on until every first choice has been used.

You can also use a table, with the first choices down the side and the second choices across the top. Each cell is one combination. A table gets full marks too, as long as every cell is filled in correctly.

Worked example 1: Listing combinations

A pizza has one base and one topping. The base is thin (T) or deep (D). The topping is mushroom (M), pepper (P) or ham (H). List all the possible combinations.

Keep the thin base fixed and go through the toppings in order: TM, TP, TH.
Your turn. Now keep the deep base fixed. How many combinations use the deep base?

Your answer

The deep base goes with each of the 3 toppings: DM, DP, DH.
Check with a table: 2 bases down the side and 3 toppings across the top make 2 \times 3 = 6 cells, and there are 6 combinations in the list.
Answer: TM, TP, TH, DM, DP, DH.

Check how many there should be

Before you finish a list, check you have them all. Think of the table, with the first choices down the side and the second choices across the top. Each row has one cell for every second choice. So 3 starters and 4 mains make 3 rows of 4 cells, which is 3 \times 4 = 12 combinations. Your list should have 12 in it.

Try it: the slider labelled First-stage choices sets the number of starters (the rows). The slider labelled Second-stage choices sets the number of mains (the columns). In each cell, the first number is the starter and the second is the main, so 2,3 means starter 2 with main 3. However you set the sliders, the number of cells is rows times columns.

This count is only a check. When a question asks you to list the combinations, or asks how many there are, write them all down: the count tells you how many to look for.

Worked example 2: Listing orders

Priya (P), Kofi (K) and Lena (L) are a relay team. Each of them runs one leg of the race: the 1st leg, the 2nd leg or the 3rd leg. One possible order is PKL. List all the possible orders they could run in.

Fix who runs the 1st leg. With Priya first, Kofi and Lena can go either way round: PKL, PLK.
Your turn. Now Kofi runs the 1st leg. How many orders start with Kofi?

Your answer

With Kofi first, Priya and Lena can go either way round: KPL, KLP.
Last, Lena runs the 1st leg: LPK, LKP. A table keeps the orders in step.
Check: each runner starts 2 orders, so there are 2 + 2 + 2 = 6 orders, and no order is written twice.
Answer: PKL, PLK, KPL, KLP, LPK, LKP.

Making numbers from cards

To make two-digit numbers from cards, choose the first digit, then go through every card that is left for the second digit. Then change the first digit. With the cards 1, 5 and 8, you get 15, 18, 51, 58, 81 and 85

Each card can be used only once, unless the question says a digit can be used more than once. Then numbers such as 55 are allowed too, and the digits 1, 5 and 8 make 9 two-digit numbers instead of 6: 11, 15, 18, 51, 55, 58, 81, 85, 88

When all three cards are used, there are 6 orders. Fix the first card, then swap the other two: 158, 185, 518, 581, 815, 851. Any three things can go in a row in 6 ways, such as the three runners in worked example 2.

A number can't start with 0. If one of the cards is a 0, leave out the numbers that begin with it. Some questions then ask which of your numbers are odd, even, multiples of 5 or prime: make the full list first, then check each number.

Worked example 3: Two-digit numbers from cards

Here are three cards: 4, 7 and 9. Ali uses two of the cards to make a two-digit number. (a) Write down all the two-digit numbers Ali can make. (b) How many of these numbers are odd?

(a) Start with 4 as the first digit. The second digit can be 7 or 9: 47, 49
Your turn. Now use 7 as the first digit. Write down the two numbers Ali can make.

Your answer

Give every value, separated by commas

With 7 first, the second digit is 4 or 9: 74, 79
With 9 first: 94, 97. That makes 3 \times 2 = 6 numbers, because each first digit has 2 cards left to go with it.
(b) A number is odd when its last digit is odd. The odd numbers end in 7 or 9: 47, 49, 79, 97
Answer: (a) 47, 49, 74, 79, 94, 97 (b) 4

Three choices in a row

When a coin is thrown three times, each outcome is three letters, one for each throw, such as HTH. The order matters: HTH and HHT are different outcomes, even though both have two heads.

Fill in the first letter for the top half and the bottom half, then the second letter in pairs, then let the third letter go H, T, H, T. Each extra throw doubles the number of outcomes: 2, then 4, then 8

Worked example 4: Outcomes of three choices

A quiz has three questions. Each answer is true (T) or false (F). Sam guesses every answer. One way is TFF: true, then false, then false. (a) List all the possible ways Sam could answer the quiz. (b) In how many of these ways does Sam answer exactly two questions true?

(a) Start with the ways that begin with T. The last two letters go TT, TF, FT, FF: TTT, TTF, TFT, TFF.
Your turn. How many ways begin with F?

Your answer

The same four endings follow F: FTT, FTF, FFT, FFF.
That makes 4 + 4 = 8 ways. Check: each question doubles the number of ways, 2, then 4, then 8
(b) Exactly two Ts: TTF, TFT and FTT. TTT has three Ts, so it doesn't count.
Answer: (a) TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF (b) 3

Worked example 5: A table of scores

In a game, Kim takes one ball from bag A and one ball from bag B. The balls in bag A are numbered 1, 2 and 5. The balls in bag B are numbered 3, 4 and 6. Kim adds the two numbers to get her score. (a) Draw a table to show all the possible scores. (b) In how many of the outcomes is the score an even number?

(a) Put bag A's numbers down the side and bag B's numbers across the top. Each cell is one outcome and holds its score. When the ball from bag A is 1: 1 + 3 = 4, 1 + 4 = 5, 1 + 6 = 7
Your turn. Now the ball from bag A is 2. Write down the three scores in this row.

Your answer

Give every value, separated by commas

2 + 3 = 5, 2 + 4 = 6, 2 + 6 = 8
When the ball from bag A is 5: 5 + 3 = 8, 5 + 4 = 9, 5 + 6 = 11. That fills all 3 \times 3 = 9 cells.
(b) The even scores are 4, 6, 8 and 8. The score 8 is in two cells, from 2 + 6 and from 5 + 3. They are different outcomes, so count both.
Answer: (a) the table, with scores 4, 5, 7, then 5, 6, 8, then 8, 9, 11 (b) 4

When the order does not matter

Sometimes the order doesn't matter. If you choose two fruits, an apple and a pear is the same choice as a pear and an apple. So each pair goes in the list only once.

Call four things P, Q, R and S. Pair P with each letter after it: PQ, PR, PS. Then pair Q with each letter after it: QR, QS. Then RS. That is 3 + 2 + 1 = 6 pairs. Never go back to an earlier letter: QP would repeat PQ.

Sometimes both things can be the same, such as two plain chocolates from a box with lots of each kind. Then pair each thing with itself as well as with each thing after it. With P, Q and R that gives PP, PQ, PR, QQ, QR, RR: 3 + 2 + 1 = 6 pairs.

Read the question carefully. When you make two-digit numbers, the order does matter: 25 and 52 are different numbers.

Worked example 6: Pairs that can be the same

An ice cream van sells three flavours: vanilla (V), mint (M) and strawberry (S). Leo buys a cone with two scoops. He can have two scoops of the same flavour. Write down all the possible combinations of flavours Leo can have.

Two scoops of the same flavour count, such as VV. The order doesn't matter: VM is the same as MV, so write each pair only once.
Start with V and pair it with every flavour, itself included: VV, VM, VS.
Your turn. Now start with M. Don't go back to V, because MV is the same as VM. How many combinations start with M?

Your answer

M pairs with itself and with the flavour after it: MM, MS.
Last, S pairs only with itself: SS. That is 3 + 2 + 1 = 6 combinations.
Answer: VV, VM, VS, MM, MS, SS.

Common mistakes

Listing in a random order. Fix one choice and go through all the others in order, so nothing is missed or repeated.
Not checking how many there should be. Multiply the numbers of choices, such as 3 \times 3 = 9, and count your list.
Writing the same pair twice. When the order doesn't matter, JM and MJ are the same combination, so write it once.
Leaving out a letter. For three throws of a coin, every outcome has three letters, such as HTT.
Counting heads instead of listing orders. Two heads and a tail is three different outcomes: HHT, HTH and THH.
Running the groups together. Keep each combination clearly separate, with commas between groups or each group in brackets.