The Product Rule for Counting - Worksheets, Questions and Revision

20 original exam-style questions - 7 pages of questions with a full mark scheme - free printable PDF.

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6.3 The Product Rule for Counting

EDEXCEL 1MA1 · Calculator allowed · about 90 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
A cafe sells sandwiches with a choice of 4 types of bread and 3 types of filling. Each sandwich has exactly one type of bread and one filling.
Work out the total number of different sandwiches possible.
(Total for Question 1 is 2 marks)
2
Priya is choosing a meal at a restaurant. The menu has 3 starters, 5 main courses and 2 desserts. She chooses exactly one starter, one main course and one dessert.
(a)Work out the number of different three-course meals Priya could choose.(2)
(b)The restaurant adds one extra starter to the menu. Work out the new number of different three-course meals possible.(1)
(Total for Question 2 is 3 marks)
3
A phone case company lets a customer choose a case colour (6 choices), a pattern (4 choices) and a strap (2 choices). Every combination of colour, pattern and strap is available.
How many different phone cases can be made?
  • A) 12
  • B) 24
  • C) 48
  • D) 96
(Total for Question 3 is 1 mark)
4
A suitcase has a combination lock with 3 dials. Each dial can be set to any digit from 0 to 9.
(a)Work out the total number of possible codes if digits can be repeated.(2)
(b)Work out the total number of possible codes if no digit can be repeated.(2)
(Total for Question 4 is 4 marks)
5
A car number plate is made from 2 letters followed by 2 digits, for example AB12. There are 26 possible letters and 10 possible digits, and letters or digits may be repeated.
Work out the total number of different number plates possible.
(Total for Question 5 is 3 marks)
6
A PIN code has 4 digits, each from 0 to 9. The first digit cannot be 0, so that the PIN does not start with 0, but digits may otherwise repeat.
Work out the number of possible PIN codes.
(Total for Question 6 is 2 marks)
7
A small vending machine sells 2 types of drink and 3 types of snack. Each purchase is one drink and one snack.
(a)By listing all the possible outcomes systematically, show that there are 6 different purchases possible.(2)
(b)A second vending machine sells 5 types of drink and 8 types of snack. Use the product rule to work out the number of different purchases possible from this machine.(2)
(Total for Question 7 is 4 marks)
8
Five different novels are placed in a row on a bookshelf.
Work out the number of different ways the five novels can be arranged.
(Total for Question 8 is 3 marks)
9
A school locker uses a 5-character code. The first two characters are different letters, chosen from the 26 letters of the alphabet, and the last three characters are different digits, chosen from 0 to 9. No letter is repeated and no digit is repeated.
(a)Work out the number of ways to choose the two letters, in order.(2)
(b)Work out the number of ways to choose the three digits, in order.(2)
(c)Hence work out the total number of different locker codes possible.(1)
(Total for Question 9 is 5 marks)
10
The digits 1, 2, 3, 4 and 5 are each used once to form a three-digit number. For example, 231 is a possible number.
Work out how many of these three-digit numbers are even.
(Total for Question 10 is 3 marks)
11
A 4-digit PIN is chosen at random, using digits 0 to 9. Digits may be repeated. Sam tries to guess the PIN in one attempt.
(a)Work out the total number of possible PINs.(2)
(b)Work out the probability that Sam guesses the correct PIN on his first attempt. Give your answer as a fraction.(2)
(Total for Question 11 is 4 marks)
12
The word MATHS has 5 different letters.
(a)Work out the number of different ways the 5 letters can be arranged.(2)
(b)Work out the number of these arrangements that begin with the letter A.(2)
(Total for Question 12 is 4 marks)
13
A shoe shop sells trainers in 4 styles and 6 colours. Every style is available in every colour, except that style D is not available in gold or in silver.
(a)Work out how many style-and-colour combinations would be possible if every style were available in every colour.(1)
(b)Work out the actual number of style-and-colour combinations available, given that style D is not available in gold or in silver.(3)
(Total for Question 13 is 4 marks)
14
There are 3 different roads from Ashby to Barton, and 4 different roads from Barton to Corby. A journey from Ashby to Corby goes through Barton.
(a)Work out the number of different routes from Ashby to Corby.(2)
(b)On the return journey from Corby to Ashby (through Barton), the driver is not allowed to use either road that was used on the way there. Work out the number of different ways to make the return journey.(2)
(Total for Question 14 is 4 marks)
15
In a race there are 8 runners. A gold, a silver and a bronze medal are awarded to the first, second and third runners respectively. No runner can win more than one medal.
Work out the number of different ways the three medals can be awarded.
(Total for Question 15 is 4 marks)
16
A restaurant offers a set menu with 4 starters and 5 main courses. Two of the starters are soups. There is 1 fish main course, and soup cannot be chosen together with the fish main course.
(a)Work out the number of starter-and-main combinations there would be with no restriction.(1)
(b)Work out the number of starter-and-main combinations that are not allowed, because of the soup-and-fish restriction.(1)
(c)Hence work out the number of valid starter-and-main combinations.(2)
(Total for Question 16 is 4 marks)
17
The digits 1, 2, 3, 4, 5, 6 and 7 are each used at most once to form a four-digit number, so no digit is repeated.
Work out how many of these four-digit numbers are even.
(Total for Question 17 is 3 marks)
18
A password is made using 6 different uppercase letters chosen from the 26 letters of the alphabet. No letter is repeated, and the order of the letters matters.
Show that there are more than one million possible passwords.
(Total for Question 18 is 3 marks)
19
A code is formed by choosing 2 different symbols, in order, from a set of n available symbols, where n ≥ 2. No symbol is repeated. The number of possible 2-symbol codes is 132.
Work out the value of n.
(Total for Question 19 is 5 marks)
20
Four different keys are laid out in a row on a table: a house key, a car key, a shed key and a bike key.
(a)Work out the total number of different ways the four keys can be arranged in a row.(2)
(b)Work out the number of these arrangements in which the house key and the car key are next to each other.(2)
(c)Hence find the probability that, if the four keys are arranged in a random order, the house key and the car key are next to each other. Give your answer as a fraction in its simplest form.(2)
(Total for Question 20 is 6 marks)
Mark scheme · 6.3 The Product Rule for Counting

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20