What is P6: Venn diagrams and tables?

Systematic enumeration means listing every permitted outcome once in a clear order.

How many marks are P6 questions worth?

In verified past-paper appearances, P6 questions on Edexcel GCSE Maths 1MA1 have been worth 2–6 marks.

Which past exam papers has P6 appeared in?

P6 has appeared in 16 verified past-paper questions indexed on this page, with direct links to the official question paper and mark scheme for each.

Enumerate sets and combinations of sets systematically, using tables, grids, Venn diagrams and tree diagrams

Venn diagrams and tables

Worked answers, methods and verified real exam appearances for P6 on Edexcel GCSE Maths 1MA1.

Explanation

Systematic enumeration means listing every permitted outcome once in a clear order.
Use a table or grid when two quantities vary, a tree for successive choices, and a Venn diagram for overlapping sets.
For a two-set Venn diagram, fill the intersection first, then the regions belonging only to each set, then the outside region.
A list of possible outcomes does not automatically make them equally likely; probability still depends on how the experiment works.

Worked example

List the numbers from $1$ to $20$ that are multiples of $3$ or $5$ by separating the intersection and the two 'only' regions.

1.Both: $\{15\}$ .
2.Multiples of $3$ only: $\{3,6,9,12,18\}$ .
3.Multiples of $5$ only: $\{5,10,20\}$ .

Answer: $\{3,5,6,9,10,12,15,18,20\}$ .

Common mistakes

Don't fall into the trap of missing an outcome or listing the same outcome twice.
Don't fall into the trap of placing intersection values in both 'only' regions as well.

Exam tip

Choose a fixed order and count your final outcomes as a check.

Worked practice

Tier 1 · Easy

A uniform is made from one of two shirts, blue or white, and one of three ties, red, silver or green. List all possible shirt-and-tie combinations.

(2)

(Total for Question 1 is 2 marks)

Mark scheme

Mark scheme for question 1
Question	Answer	Mark	Mark scheme
1	Blue-red, blue-silver, blue-green, white-red, white-silver, white-green.	2	Hold the shirt colour fixed and list every tie, then repeat for the other shirt. This gives $2\times3=6$ combinations, each appearing once.

Tier 2 · Standard

The universal set is the integers from

1

20

. Set A contains the multiples of

3

and set B contains the factors of

18

. Enumerate the four regions of a Venn diagram for A and B.

(4)

(Total for Question 2 is 4 marks)

Mark scheme

Mark scheme for question 2
Question	Answer	Mark	Mark scheme
2	$A\cap B=\{3,6,9,18\}$ . A only $=\{12,15\}$ ; B only $=\{1,2\}$ . Neither $=\{4,5,7,8,10,11,13,14,16,17,19,20\}$ .	4	List $A=\{3,6,9,12,15,18\}$ and $B=\{1,2,3,6,9,18\}$ . Put their common values in the intersection, remove these to find each 'only' region, then place every unused integer from $1$ to $20$ outside both circles.

Tier 3 · Hard

A three-digit number is formed from three different digits chosen from

1

2

3

and

4

. Enumerate all the numbers that are greater than

230

and even. How many of these numbers do not contain the digit

1

(4)

(Total for Question 3 is 4 marks)

Mark scheme

Mark scheme for question 3
Question	Answer	Mark	Mark scheme
3	$234,312,314,324,342,412,432$ . $4$ of these numbers do not contain the digit $1$ .	4	Work systematically by hundreds digit. Starting with $2$ gives $234$ ; starting with $3$ gives $312,314,324,342$ ; starting with $4$ gives $412,432$ . The values without digit $1$ are $234,324,342,432$ , so there are $4$ .

Tier 1 · Easy

Write down all the ordered pairs of positive integers that have a total of

7

and have the first number less than the second number.

(2)

(Total for Question 4 is 2 marks)

Mark scheme

Mark scheme for question 4
Question	Answer	Mark	Mark scheme
4	$(1,6),(2,5),(3,4)$	2	Start with first number $1$ and increase it systematically: $(1,6),(2,5),(3,4)$ . The next pair would have the first number greater than the second, so the list is complete.

Tier 2 · Standard

A journey uses either a bus or a train, followed by walking, cycling or taking a taxi. A bus journey cannot be followed by a taxi. List all the possible journeys and write down how many there are.

(3)

(Total for Question 5 is 3 marks)

Mark scheme

Mark scheme for question 5
Question	Answer	Mark	Mark scheme
5	Bus-walk, bus-cycle, train-walk, train-cycle, train-taxi. $5$ journeys.	3	List every second stage after bus, omitting taxi, then list every second stage after train. This gives $2+3=5$ permitted journeys.

Tier 3 · Hard

The universal set is the integers from

1

30

. Set A contains multiples of

2

, set B contains square numbers, and set C contains factors of

24

. List the integers in each region that is in exactly two of the three sets and work out how many integers are in exactly two sets.

(4)

(Total for Question 6 is 4 marks)

Mark scheme

Mark scheme for question 6
Question	Answer	Mark	Mark scheme
6	A and B only: $\{16\}$ ; A and C only: $\{2,6,8,12,24\}$ ; B and C only: $\{1\}$ . $7$ integers are in exactly two sets.	4	The only number in all three sets is $4$ , so exclude it from the exactly-two regions. The remaining common values are $16$ in A and B, $2,6,8,12,24$ in A and C, and $1$ in B and C. There are $1+5+1=7$ integers.

Tier 2 · Standard

The first coordinate of an ordered pair is chosen from

\{-2,1,4\}

and the second coordinate is chosen from

\{-1,3\}

. List all the possible ordered pairs. Then list the pairs for which the sum of the coordinates is greater than

2

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
Question	Answer	Mark	Mark scheme
7	$(-2,-1),(-2,3),(1,-1),(1,3),(4,-1),(4,3)$ . Sum greater than $2$ : $(1,3),(4,-1),(4,3)$ .	3	Fix the first coordinate at $-2$ , then $1$ , then $4$ , pairing each with both possible second coordinates. Check the coordinate sums; only $1+3$ , $4+(-1)$ and $4+3$ are greater than $2$ .

Tier 3 · Hard

A ticket code consists of one letter, A, B or C, followed by two different digits chosen from

1

2

3

and

4

. For an A-code, the final digit must be even. For a B-code, the two digits must have a total of

5

. For a C-code, the first digit must be less than the second digit. List every permitted code and work out the total number of codes.

(4)

(Total for Question 8 is 4 marks)

Mark scheme

Mark scheme for question 8
Question	Answer	Mark	Mark scheme
8	A-codes: A₁₂, A₃₂, A₄₂, A₁₄, A₂₄, A₃₄. B-codes: B₁₄, B₄₁, B₂₃, B₃₂. C-codes: C₁₂, C₁₃, C₁₄, C₂₃, C₂₄, C₃₄. $16$ permitted codes.	4	For A, fix the last digit as $2$ or $4$ and choose any different first digit, giving $6$ codes. The ordered digit pairs totalling $5$ are $14,41,23,32$ , giving $4$ B-codes. The increasing pairs are $12,13,14,23,24,34$ , giving $6$ C-codes. The total is $6+4+6=16$ .

Tier 3 · Hard

A robot makes exactly five moves. Each move is either one square east, E, or one square north, N. It must make three east moves and two north moves, but it must not make two north moves consecutively. List every permitted route and work out how many there are.

(3)

(Total for Question 9 is 3 marks)

Mark scheme

Mark scheme for question 9
Question	Answer	Mark	Mark scheme
9	NENEE, NEENE, NEEEN, ENENE, ENEEN, EENEN; $6$ routes.	3	Place the two N moves systematically in non-adjacent positions among the five moves, then fill the other positions with E. The possible N-position pairs are $(1,3),(1,4),(1,5),(2,4),(2,5),(3,5)$ , giving the six listed routes and no repeats.

Q10

Tier 3 · Hard

A club chooses a captain and a deputy from students A, B, C, D and E. The same student cannot hold both roles. Student C cannot be captain, and students A and B cannot both hold a role. List every possible ordered captain-deputy pair and work out the total number of choices.

(5)

(Total for Question 10 is 5 marks)

Mark scheme

Mark scheme for question 10
Question	Answer	Mark	Mark scheme
10	AC, AD, AE; BC, BD, BE; DA, DB, DC, DE; EA, EB, EC, ED. $14$ choices.	5	List by captain. With A as captain, B is excluded and the deputy can be C, D or E; similarly there are three choices with B as captain. With D or E as captain, any of the other four students can be deputy because A and B are not then serving together. This gives $3+3+4+4=14$ .

Verified exam appearances

Series	Paper	Question	Marks	Calculator	Tier	Links
2023-06	1F	Q24	5	Non-calculator	Foundation	QP MS
2024-06	3F	Q27	4	Allowed	Foundation	QP MS
2024-06	3H	Q6	4	Allowed	Higher	QP MS
2022-06	2H	Q16	6	Allowed	Higher	QP MS
2021-11	1H	Q2	3	Non-calculator	Higher	QP MS
2022-11	3H	Q6	4	Allowed	Higher	QP MS
2021-11	1F	Q21	3	Non-calculator	Foundation	QP MS
2024-11	1F	Q11	5	Non-calculator	Foundation	QP MS
2019-06	3H	Q1	5	Allowed	Higher	QP MS
2019-06	3F	Q24	5	Allowed	Foundation	QP MS
2024-06	1H	Q4	3	Non-calculator	Higher	QP MS
2019-11	3F	Q20	4	Allowed	Foundation	QP MS
2024-11	2F	Q23	2	Allowed	Foundation	QP MS
2023-11	2H	Q21	4	Allowed	Higher	QP MS
2024-06	1F	Q23	3	Non-calculator	Foundation	QP MS
2024-11	2H	Q4	2	Allowed	Higher	QP MS

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