The Product Rule for Counting worksheet

10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.

1. 4 marks

   A café offers a set menu with 4 starters, 6 main courses and 5 desserts.

   (a)

   Rosa chooses one starter, one main course and one dessert.

   Work out the number of different meals she could choose.

   (b)

   A second customer wants a starter and a main course, and may or may not have a dessert.

   Work out the number of different meals this customer could choose.

2. 4 marks

   A password is made up of 3 letters followed by 2 digits.

   Each letter can be any of the 26 letters of the alphabet.

   Each digit can be any of the digits from 1 to 4.

   Letters may be repeated and digits may be repeated.

   (a)

   Calculate how many different passwords can be made.

   (b)

   Calculate how many of the passwords have all of their digits odd.

3. 5 marks

   There are 2 roads from Ashford to Barton, 4 roads from Barton to Colwick and 3 roads from Colwick to Denby.

   There are no other roads between these towns.

   (a)

   Calculate the number of different routes from Ashford to Denby.

   (b)

   A driver goes from Ashford to Denby and then back again.

   On the way back the driver must not use any road already used on the way out.

   Calculate the number of different there-and-back journeys.

4. 4 marks

   A drama society has 9 members.

   Three of the members are to be chosen: one as team leader, one as note-taker and one as timekeeper.

   (a)

   No member is allowed to hold more than one of the three posts.

   Calculate the number of different ways the three posts can be filled.

   (b)

   Suppose instead that a member is allowed to hold more than one of the three posts.

   Calculate the number of different ways the three posts can be filled now.

5. 5 marks

   Here are six digit cards.

   \(9,\ 5,\ 8,\ 1,\ 6,\ 2\)

   Grace uses three of the cards to make a three-digit number. No card can be used more than once in a number.

   (a)

   Calculate how many different three-digit numbers she can make.

   (b)

   Calculate how many of these three-digit numbers are odd.

6. 4 marks

   A café offers a set menu with 3 starters, 4 main courses and 6 desserts.

   (a)

   Tomas chooses one starter, one main course and one dessert.

   Find the number of different meals he could choose.

   (b)

   A second customer wants a starter and a main course, and may or may not have a dessert.

   Find the number of different meals this customer could choose.

7. 4 marks

   A luggage-locker code is made up of 2 letters followed by 3 digits.

   Each letter can be any of the 26 letters of the alphabet.

   Each digit can be any of the digits from 1 to 9.

   Letters may be repeated and digits may be repeated.

   (a)

   Calculate how many different codes can be made.

   (b)

   Calculate how many of the codes have all of their digits odd.

8. 5 marks

   There are 3 roads from Ravenden to Selby, 5 roads from Selby to Thorne and 2 roads from Thorne to Upton.

   There are no other roads between these towns.

   (a)

   Calculate the number of different routes from Ravenden to Upton.

   (b)

   A driver goes from Ravenden to Upton and then back again.

   On the way back the driver must not use any road already used on the way out.

   Calculate the number of different there-and-back journeys.

9. 4 marks

   A photography club has 16 members.

   Three of the members are to be chosen: one as editor, one as photographer and one as designer.

   (a)

   No member is allowed to hold more than one of the three posts.

   Find the number of different ways the three posts can be filled.

   (b)

   Suppose instead that a member is allowed to hold more than one of the three posts.

   Find the number of different ways the three posts can be filled now.

10. 5 marks

    Here are five digit cards.

    \(6,\ 5,\ 1,\ 2,\ 7\)

    Owen uses three of the cards to make a three-digit number. No card can be used more than once in a number.

    (a)

    Work out how many different three-digit numbers he can make.

    (b)

    Work out how many of these three-digit numbers are odd.

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