Factorials and arrangements Cambridge IGCSE Additional Mathematics revision
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In plain words
How many ways can five books be lined up on a shelf? Five choices for the first place, four for the next, then three, two, one: 5 × 4 × 3 × 2 × 1 = 120. Counting like this, where the order matters, is the subject of permutations.
6 things to know
- n! (n factorial) is n × (n − 1) × (n − 2) × … × 1. By definition, 0! = 1.
- n different objects can be arranged in a line in n! ways.
- A permutation is an arrangement: the order matters. The number of ways of arranging r objects chosen from n is n! ÷ (n − r)!.
- That is just n × (n − 1) × … for r numbers: choosing and arranging 3 from 7 is 7 × 6 × 5.
- If certain objects must be together, treat them as one block. Arrange the blocks, then multiply by the number of ways of arranging the objects inside the block.
- If some positions are restricted, such as an even last digit, fill those positions first.
Worked example
In how many ways can 6 people stand in a line if two of them, Ali and Bo, must be next to each other?
- Treat Ali and Bo as one block. There are now 5 things to arrange: 5! = 120 ways.
- Inside the block, Ali and Bo can be in 2 orders.
- 120 × 2 = 240 ways.
Worked example
How many even four-digit numbers can be made from the digits 1, 2, 3, 4, 5 and 6, with no digit repeated?
- The last digit must be even: 2, 4 or 6. That is 3 choices.
- The other three places are filled from the 5 digits left: 5 × 4 × 3 = 60 ways.
- 3 × 60 = 180 numbers.
Tips and tricks
- Deal with the restriction first, then fill the rest.
- Use a permutation when the order matters: places in a race, digits in a number, letters in a word.
It lands in your notebook with its questions as flashcards.
Factorials and arrangements: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What is 5!?
5 × 4 × 3 × 2 × 1.
What is 0!?
By definition.
In how many ways can 4 different books be arranged on a shelf?
4!.
In how many ways can 2 of 6 people be chosen as chair and secretary?
6 × 5: the two jobs are different, so order matters.
When does a problem need permutations and not combinations?
An arrangement is a permutation.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 8 marks. Write your answers on paper, then check them.
Factorials and arrangements
Cambridge IGCSE Additional Mathematics 0606 · 8 marks · papermunch.org
Name ______________________________ Date ______________
Evaluate 6! and the number of permutations of 2 objects chosen from 8.[2]
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720 and 56. 8 × 7 = 56.
Find the number of ways in which the first three places can be filled in a race with 5 runners.[3]
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60. 5 × 4 × 3.
Find the number of arrangements of the letters of the word PLANET in which the two vowels are next to each other.[3]
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240. The vowels form one block: 5! × 2.
Answers: Factorials and arrangements
- 1. 720 and 56. 8 × 7 = 56.
- 2. 60. 5 × 4 × 3.
- 3. 240. The vowels form one block: 5! × 2.