New
New
Year 11
Higher

Combinations

I can derive the product rule for counting, including when to divide by 2.

New
New
Year 11
Higher

Combinations

I can derive the product rule for counting, including when to divide by 2.

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Lesson details

Key learning points

  1. By systematically listing the outcomes for two or more events, a pattern begins to emerge
  2. This pattern can be generalised so that counting possible outcome combinations is quicker
  3. Whether the order of the outcomes matters affects the number of possible outcomes
  4. If order does not matter then repetition can be accounted for when considering the total number of outcomes

Keywords

  • Trial - A trial is a single, predefined test.

  • Outcome - An outcome is a result of a trial.

  • Systematically - When listing outcomes systematically, they are listed in such a way as to ensure all outcomes are recorded.

Common misconception

When using the product rule, pupils may be unsure whether they should divide the product by 2.

Pay extra attention to the context of the problem. If the order of the two outcomes is irrelevant, then you should divide by two. If each of the two outcomes is allocated to a separate event (e.g. 1st and 2nd) then do not divide by 2.

Some of the examples in the lesson could be enacted to help pupils to visualise what is happening.
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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6 Questions

Q1.
Two spinners are spun once each. The table shows the outcomes for each spinner. Match each value from the frequency tree to its value.
An image in a quiz
Correct Answer:$$a$$,24

24

Correct Answer:$$c$$,15

15

Correct Answer:$$e$$,3

3

Correct Answer:$$g$$,5

5

Q2.
An investigation into some rhubarb plants aims to see if using a special fertiliser (F) increases the chance of the plant yielding rhubarb (R). The diagrams show the results. $$b$$ = .
An image in a quiz
Correct Answer: 195
Q3.
An investigation into some rhubarb plants aims to see if using a special fertiliser (F) increases the chance of the plant yielding rhubarb (R). The diagrams show the results. $$c$$ = .
An image in a quiz
Correct Answer: 240
Q4.
An investigation across 60 days explores whether days with rain (R) were more likely to see deer (D) in urban areas. The value of $$y$$ is .
An image in a quiz
Correct Answer: 4
Q5.
An investigation across 60 days explores whether days with rain (R) were more likely to see deer (D) in urban areas. The value of $$z$$ is .
An image in a quiz
Correct Answer: 31
Q6.
A game developer tested to see whether a playable character was more likely to win (W) with a rebalance (R) of the character’s stats or not. P(R'W) = .
An image in a quiz
Correct Answer: 0.12

6 Questions

Q1.
There are 4 children sat around one table and 8 children sat around another table. The teacher chooses 1 child from each table. In how many different ways could the teacher do this?
2
12
24
Correct answer: 32
64
Q2.
Here is a restaurant menu for three courses. A customer can choose one starter, one main course and one dessert in different ways.
An image in a quiz
Correct Answer: 24
Q3.
The restaurant runs out of tiramisu. A customer can choose one starter, one main course and one dessert in different ways now.
An image in a quiz
Correct Answer: 12, twelve
Q4.
Alex has a pack of 5 cards. He deals one card to himself and deals one card to Sam. In how many different ways can this happen?
An image in a quiz
9
10
Correct answer: 20
25
40
Q5.
Izzy has a pack of 7 cards. She takes 2 cards at random from the pack. In how many different ways can this happen?
An image in a quiz
13
14
Correct answer: 21
42
Q6.
Alex draws 2 different cards from set A and 2 different cards from set B. There are different ways of doing this.
An image in a quiz
Correct Answer: 126