Counting Principles (AQA GCSE Further Maths) : Revision Note

Product Rule for Counting

What is meant by counting principles?

• Counting principles state that if there are m ways to do one thing and n ways to do another there are m × n ways to do both things

• Applying counting principles allows us to ...

  • ... see patterns in real world situations

  • ... find the number of combinations or arrangements of a number of items

  • ... find the number of ways of choosing some items from a list of items

• When you have a question like “How many ways…?” you should always look for the words “AND” and “OR”

  • “AND means ×” 
    “OR means +”  

How do I choose an item from a list of items AND another item from a different list of items?

• If a question requires you to choose an item from one list AND an item from another list you should multiply the number of options in each list

  • In general if you see the word 'AND' you will most likely need to 'MULTIPLY'

• For example if you are choosing a pen and a pencil from 4 pens and 5 pencils:

  • You can choose 1 item from 4 pens AND 1 item from 5 pencils

  • You will have 4 × 5 different options to choose from

How do I choose an item from a list of items OR another item from a different list of items?

• If a question requires you to choose an item from one list OR an item from another list you should add the number of options in each list

  • In general if you see the word 'OR' you will most likely need to 'ADD'

• For example if you are choosing a pen or a pencil from 4 pens and 5 pencils:

  • You can choose 1 item from 4 pens OR 1 item from 5 pencils

  • You will have 4 + 5 different options to choose from

How many ways can n different objects be arranged?

• When considering how many ways you can arrange a number of different objects in a row it’s a good idea to think of how many of the objects can go in the first position, how many can go in the second and so on

• For there are two options for the first position and then there will only be one object left to go in the second position so

  • To arrange the letters A and B we have

    • AB and BA

  • For   there are three options for the first position and then there will be two objects for the second position and one left to go in the third position so

    • To arrange the letters A, B and C we have

    • For n objects there are  options for the first position,  options for the second position and so on until there is only one object left to go in final position

    • The number of ways of arranging different objects is  

• If the objects being arranged can be repeated in the list then we do not reduce each term by one

  • For example to arrange the letters A, B and C where each letter can be used more than once we have 

    • AAA, AAB, ABB, ...

    • There are three options for the first position, three options for the second and three for the third

    • There are 3 × 3 × 3 = 27 possibilities

  • For example to arrange any 4 out of the 26 letters of the English alphabet, where each letter can be used more than once

    • There are 26 options for the first position, 26 for the second, 26 for the third and 26 for the fourth

    • There are 26 × 26 × 26 × 26 = 26⁴ possibilities