Combinations

Topic

Higher June 2025 Paper 2 Q14

EdexcelCurrent spec3 marksCombinations

14 There are 15 dogs in a dog show.
One dog is awarded first prize.
A different dog is awarded second prize.

There are 12 cats in a cat show.
One cat is awarded first prize.
A different cat is awarded second prize.

Work out how many different ways these four prizes can be awarded. (3)

Mark scheme

Answer	Mark	Mark scheme
27 720	P1	for \(15 \times 14\ (= 210)\) or \(12 \times 11\ (= 132)\) or \(15 \times 12\ (= 180)\)
	P1	for a complete method eg \(15 \times 14 \times 12 \times 11\)
	A1	cao SCB1 for an answer of 342 if P0 awarded

Additional guidance

If values given as probabilities can award P marks only

If a correct product seen and divided (by 2 or 4 or 24) award P marks

Foundation June 2025 Paper 3 Q8

EdexcelCurrent spec2 marksCombinations

8 Here is a menu.

Drink	Cake
Coffee (C)	Brownie (B)
Tea (T)	Flapjack (F)
Lemonade (L)	Scone (S)

Asma can choose one drink and one cake.

Write down all the possible combinations Asma can choose. (2)

Mark scheme

Answer	Mark	Mark scheme
CB, CF, CS, TB, TF, TS, LB, LF, LS	B2	for fully correct list with no repeats
CB, CF, CS, TB, TF, TS, LB, LF, LS	(B1	for at least 4 correct combinations, condone repeats and incorrect combinations)

Additional guidance

Accept unambiguous abbreviations
Accept CB or BC etc

Higher November 2024 Paper 2 Q14

EdexcelCurrent spec2 marksCombinations

14 There are 10 football teams in a league.
Each team plays every other team 4 times.

Work out the total number of games played. (2)

Mark scheme

Answer	Mark	Mark scheme
180	M1	for \(10 \times 9 \times 4\ (= 360)\) or \(10 \times 9 \times 4 \div 2\) or \(10 \times 9 \div 2\ (= 45)\) or \((9 \times 4) + (8 \times 4) + (7 \times 4) + (6 \times 4) + (5 \times 4) + (4 \times 4) + (3 \times 4) + (2 \times 4) + (1 \times 4)\) oe
180	A1	cao

Foundation June 2025 Paper 3 Q3

AQACurrent spec3 marksCombinations

(a) Jodie makes a sandwich using one type of bread and one filling.

Bread	Filling
White (W) or Brown (B)	Cheese (C) or Ham (H) or Tuna (T)

Complete the table of the 6 possible sandwiches they could make.

One has been done for you. [2 marks]

(b) What fraction of the possible sandwiches use Cheese? [1 mark]

Mark scheme (a)

Answer	Mark	Comments
WC WH WT BC BH BT	B2	any order B1 three or four correct

Additional guidance

Mark the table unless crossed out and replaced

Ignore repeats for B1

Accept words

Allow WC to be missing or written again if list restarted

Answer	Mark	Comments
\(\dfrac{2}{6}\)	B1ft	oe fraction ft their table with max 6 sandwiches from their part (a)

Foundation November 2024 Paper 2 Q9

AQACurrent spec2 marksCombinations

9 Callum is making pizza.

He can choose three different toppings from

ham (H)
sweetcorn (S)
onion (O)
mushrooms (M).

List all the possible options for the three toppings.

The first one has been done for you. [2 marks]

H S O

Mark scheme

Answer	Mark	Comments
HSO HSM HOM SOM	B2	with no errors may be given as words any order B1 for any one correct option stated

Additional guidance

Ignore repeated answers eg HSM and SHM

Repeated toppings is an error eg HHS

2 or 4 toppings is an error eg SM or HSOM

Foundation June 2024 Paper 1 Q7

AQACurrent spec3 marksCombinations Properties of Numbers

(a) At a restaurant, vegan pizzas have two different toppings.

The toppings are

sweetcorn (S) mushrooms (M) peppers (P)

Complete the table to list all the possible pairs of toppings. [1 mark]

(b) At the restaurant, dough balls can be ordered in small portions and large portions.

Small portion
6 dough balls

Large portion
10 dough balls

A group of people want to order exactly 44 dough balls.

Show how they can do this. [2 marks]

Number of Small portions \(\ldots\ldots\ldots\)
Number of Large portions \(\ldots\ldots\ldots\)

Answer	Mark	Comments
SP MP with no others	B1	oe accept in words

Mark scheme (b)

Answer	Mark	Comments
A trial of at least 3 portions involving small and large with correct total seen or 24 and 20 chosen or \(4 \times 6\) (\(= 24\)) and \(2 \times 10\) (\(= 20\))	M1	eg \(2 \times 6 + 10 = 22\) or 3S and 2L is 38
4 small and 2 large	A1

Additional guidance

Ignore incorrect trials if a correct trial or the correct answer is seen
Any unambiguous indication eg 2L 4S	M1A1
\(5 \times 6 + 2 \times 10 = 54\)	M0

Foundation November 2023 Paper 3 Q12

AQACurrent spec3 marksCombinations

12 Here are the subjects available in year 12 at a school.

Block 1	Block 2	Block 3	Block 4
Maths (M)	Geography (G)	English (E)	Spanish (S)
History (H)	Drama (D)	Physics (P)	Biology (B)
French (F)	Chemistry (C)	ICT (I)	Art (A)

Students choose three subjects.

They cannot choose more than one subject from a block.

Lian decides

to study Maths

not to study Geography, Chemistry, Physics or ICT.

By listing, show that there are seven groups of three subjects that Lian could choose. [3 marks]

Subject 1	Subject 2	Subject 3

Mark scheme

Answer	Mark	Comments
Maths Drama English (MDE) Maths Drama Spanish (MDS) Maths Drama Biology (MDB) Maths Drama Art (MDA) Maths English Spanish (MES) Maths English Biology (MEB) Maths English Art (MEA)	B3	any order of subjects and/or rows B2 5 or 6 rows correct B1 3 or 4 rows correct

Additional guidance

Accept any indication for subject

Allow repeats and additional rows for up to B2

Higher June 2017 Paper 2 Q5

AQACurrent spec3 marksCombinations

5 A code has 4 digits.

Each digit is a number from 0 to 9

Digits may be repeated.

The code starts \(\quad 5 \;\; 4 \;\; 1\)

(a) Amy knows the last digit is odd but not 7

She chooses a different odd number at random.

What is the probability that she chooses the correct number? [1 mark]

(b) The 4-digit code is changed to an even number.

The first digit is 3

How many possible codes are there? [2 marks]

Mark scheme (a)

Answer	Mark	Comments
\(\dfrac{1}{4}\) or 0.25 or 25%	B1	oe

Additional guidance

Ratio eg 1 : 4 or 1 : 3	B0
\(\dfrac{1}{4}\) seen and answer 1 : 4	B1
Expressed only in words eg 1 out of 4	B0
1 out of 4 and \(\dfrac{1}{4}\)	B1
\(\dfrac{1}{4}\) seen with change to incorrect decimal or incorrect percentage eg \(\dfrac{1}{4}\) and answer 0.4	B1
Ignore chance words if \(\dfrac{1}{4}\) seen eg \(\dfrac{1}{4}\) and answer Likely	B1

Mark scheme (b)

Answer	Mark	Comments
\((1 \times)\ 10\ (\times)\ 10\ (\times)\ 5\) or \(\dfrac{10 \times 10 \times 10}{2}\) or \(\dfrac{1000}{2}\)	M1	oe
500	A1	SC1 5 or 324 or 400 or 405

Additional guidance

\(10 + 10 + 5\)	M0A0
SCs are for the answers from not including zero at least once ie \(9 \times 9 \times 4\) or \(10 \times 10 \times 4\) or \(9 \times 9 \times 5\) or from a misread ie \(1 \times 1 \times 1 \times 5\)