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GCSE Maths · Higher · Chapter 16: Counting, accuracy, powers and surds · Lesson 5

Counting choices and outcomes

Grades 5–8About 50 minutesNo calculator41 marks
Hints online, answers free with an account:
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NameClassDate
A

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

  1. A1
    Needed today

    28 people: 17 like tea, 12 like coffee, 6 like both. How many like neither?

    [1 mark]
    Answer
  2. A2
    Last lesson

    A length is 7.9 cm to 1 d.p. Write down its lower bound.

    [1 mark]
    Answer
  3. A3
    From chapter 11

    From 28 m away the angle of elevation to the top of a tree is . How tall is the tree?

    [1 mark]
    Answer
  4. A4
    From chapter 16

    Work out as a fraction.

    [1 mark]
    Answer
B

Example, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it.

Example 1

A café has 5 sandwiches, 4 drinks and 3 snacks. A lunch is one of each. How many different lunches are there?

Hint Multiply the number of choices at each stage.
  1. B1

    A bike is built from one of 8 frames, one of 5 colours and one of 2 seats. How many different bikes are there?

    [2 marks]
    Answer
Example 2

In how many different ways can 3 of 7 different books be put in a row on a shelf?

Hint 7 choices for the first place, then one fewer each time.
  1. B2

    In how many different ways can 4 of 9 different photos be hung in a row?

    [2 marks]
    Answer
C

Practice

Grades 6–7
  1. C1

    A lock has 3 dials. Each dial shows one of the letters A to F. How many different codes are there?

    [2 marks]
    Answer
  2. C2

    In how many different orders can 5 different pens be put in a row?

    [1 mark]
    Answer
  1. C3

    A pair of students is chosen from 10 students. The order does not matter. How many different pairs are there?

    [2 marks]
    Answer
  2. C4

    Three-digit numbers are made from the digits 1 to 7. Digits may repeat. How many of the numbers are even?

    [2 marks]
    Answer
  3. C5

    How many different arrangements are there of the letters in the word TOFFEE?

    [2 marks]
    Answer
  4. C6

    A password is 1, 2 or 3 letters long. Each letter is one of A, B, C, D or E, and letters may repeat. How many different passwords are there?

    [2 marks]
    Answer
  5. C7

    A club gives each of its 600 members a code of 2 different symbols in order, such as . What is the smallest number of symbols the club needs?

    [2 marks]
    Answer
  6. C8

    Three-digit numbers are made from the digits 2 to 8. No digit is used twice. How many of the numbers are greater than 600?

    [2 marks]
    Answer
D

Exam-style questions

Grades 7–8
  1. D1

    Ella says, "A team of 3 can be chosen from 6 people in ways."

    Explain why Ella is wrong, and work out the correct number of teams.

    [2 marks]
  1. D2

    Mia makes an ice cream. She chooses one of 3 cones, two different scoops from 8 flavours, and one of 5 toppings. The order of the two scoops does not matter.

    (a)

    How many different pairs of scoops can she choose?

    [2 marks]
    Answer
    (b)

    How many different ice creams can she make?

    [2 marks]
    Answer
    (c)

    One of her scoops must be chocolate. How many different ice creams can she make now?

    [2 marks]
    Answer
    Total for D2: 6 marks
  2. D3

    A committee of 4 people is chosen from 6 teachers and 3 parents. The order of the people does not matter.

    (a)

    How many different committees are possible?

    [2 marks]
    Answer
    (b)

    How many of the committees have at least one parent?

    [3 marks]
    Answer
    (c)

    A committee is chosen at random. Work out the probability that it has no parents. Give your answer as a fraction in its simplest form.

    [2 marks]
    Answer
    Total for D3: 7 marks
E

Extension

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

  1. E1

    How many four-digit whole numbers are greater than 5000 and have four different digits?

    [3 marks]
    Hint 1 · What to try

    Start with the first digit. Which digits can it be?

    Hint 2 · The first line

    The first digit is 5, 6, 7, 8 or 9, so there are 5 choices. The number 5000 itself has repeated digits.

    Hint 3 · The full method

    The other three places use the 9 digits left, then 8, then 7: .

    Answer
F

How did it go?

Colour one circle on each line.
Not yetNearlyYes
I can use the product rule to count choices.
I can count arrangements, with and without repeats.
I can count choices when the order does not matter.

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