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The Product Rule for Counting

discia.org 10 questions · 36 marks
Name Date
  1. Either paper 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.

    Show answer
    (a) \(120\)
    (b) \(144\)
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  2. Either paper 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.

    Show answer
    (a) \(281,216\)
    (b) \(70,304\)
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  3. Either paper 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.

    Show answer
    (a) \(24\)
    (b) \(144\)
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  4. Either paper 2 marks

    Tomas is in a restaurant. There are 3 starters, 8 main courses and some desserts on the menu.

    Tomas is going to choose one starter, one main course and one dessert.

    Tomas says there are 280 ways that he can choose a starter, a main course and a dessert.

    Could Tomas be correct? You must show how you get your answer.

    Show answer
    No: \(280 \div 24 = 11\frac{2}{3}\), which is not a whole number of desserts
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  5. Either paper 2 marks

    A bakery has 7 iced buns, 11 flapjacks and 4 muffins.

    Idris is going to choose one of the following combinations: an iced bun and a muffin; or a flapjack and a muffin; or an iced bun, a flapjack and a muffin.

    Show that there are 380 different ways for Idris to choose his cakes.

    Show answer
    \(7 \times 4 + 11 \times 4 + 7 \times 11 \times 4 = 28 + 44 + 308 = 380\), so 380
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  6. Either paper 4 marks

    A chess club has 15 members.

    Three of the members are to be chosen: one as president, one as secretary and one as social secretary.

    (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.

    Show answer
    (a) \(2,730\)
    (b) \(3,375\)
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  7. Either paper 5 marks

    Here are six digit cards.

    \(6,\ 2,\ 3,\ 5,\ 7,\ 8\)

    Noah 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 he can make.

    (b)

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

    Show answer
    (a) \(120\)
    (b) \(60\)
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  8. Either paper 3 marks
    Here are two methods for making a 4-digit code. Each digit is one of the digits 0 to 9. A digit may be repeated unless its box says otherwise.
    1st digit2nd digit3rd digit4th digit
    Method Abetween 2 and 8 inclusiveoddbetween 2 and 8 inclusiveless than 6
    Method Bless than 8greater than 3any digitless than 3
    Which method gives the greater number of possible codes? You must show your working.
    Show answer
    Method A: 1,470 codes, compared with 1,440
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  9. Either paper 3 marks

    Tomas has to set a 4-digit passcode on his phone.

    Each digit of the passcode is a whole number from 1 to 9. Tomas can use each number more than once.

    Tomas tells Leah that

    the first digit is a cube number

    the second digit is a multiple of 3

    the third digit is less than 6

    the fourth digit is less than 4.

    The diagram shows one possible 4-digit passcode.

    8651

    Leah is going to have one attempt at guessing Tomas's passcode.

    Work out the probability that Leah guesses the passcode on the first attempt.

    Show answer
    \(\frac{1}{ 90 }\)
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  10. Either paper 4 marks

    A school canteen offers a set menu with 6 starters, 5 main courses and 3 desserts.

    (a)

    Omar 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.

    Show answer
    (a) \(90\)
    (b) \(120\)
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discia.org The Product Rule for Counting · sheet 1
Answers The Product Rule for Counting · sheet 1
  1. (a) \(120\)
    (b) \(144\)
  2. (a) \(281,216\)
    (b) \(70,304\)
  3. (a) \(24\)
    (b) \(144\)
  4. No: \(280 \div 24 = 11\frac{2}{3}\), which is not a whole number of desserts
  5. \(7 \times 4 + 11 \times 4 + 7 \times 11 \times 4 = 28 + 44 + 308 = 380\), so 380
  6. (a) \(2,730\)
    (b) \(3,375\)
  7. (a) \(120\)
    (b) \(60\)
  8. Method A: 1,470 codes, compared with 1,440
  9. \(\frac{1}{ 90 }\)
  10. (a) \(90\)
    (b) \(120\)
discia.org The Product Rule for Counting · sheet 1