2.2.1 Arrangements & Factorials

Arrangements

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 $n = 2$ 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 $n = 3$ 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
    - ABC, ACB, BAC, BCA, CAB and CBA
  - For n objects there are $n$ options for the first position, $n - 1$ 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 $n \times (n - 1) \times (n - 2) \times . . . \times 2 \times 1$

What happens if the objects are not all different?

Consider arranging two identical objects, although there are still two different ways you could place the objects down next to each other, the arrangements would look exactly the same

To arrange the letters A₁ and A₂ we have

A₁A₂ and A₂A₁
These are exactly the same, so there is only one way to arrange the letters A and A

To arrange the letters A₁, A₂ and C we have

A₁A₂C, A₂A₁C, A₁C A₂, , A₂C A₁, C A₁A₂, C A₂A₁
Although the two letter As were placed separately, they are identical and so each pattern has been repeated twice
There are 6 ways to arrange the letters A, A and C, but with some duplicates
There are $\frac{6}{2} = 3$ different ways to arrange the letters A, A and C

- If there are two identical objects within a group of objects to be arranged, the number of ways of arranging different objects should be divided by 2

Consider arranging three identical objects, although there are still six different ways you could place the objects down next to each other, the arrangements would look exactly the same

- To arrange the letters A₁, A₂ and A₃
  - A₁A₂ A₃ , A₁A₃A₂, A₂A₁ A₃ , A₂A₃A₁ , A₃A₁ A₂ , A₃A₂A₁
  - However, if these were all A, we would have AAA repeated six times
- To find the number of arrangements of the letters A, A, A and C we would have to consider the number of ways of arranging four letters if they were all different and then divide by the number of ways AAA is repeated
  - Four different letters could be arranged $4 \times 3 \times 2 \times 1$ = 24 ways
  - AAA would be repeated six times so we would need to divide by 6
  - There are four different ways to arrange the letters A, A, A and C
If there are three identical objects within a group of n objects to be arranged, the number of ways of arranging $n$ different objects should be divided by 6
If there are r identical objects within a group of n objects to be arranged, the number of ways of arranging $n$ different objects should be divided by the number of ways of arranging r different objects
- If there are r identical objects within a group of n objects to be arranged, the number of ways of arranging n the objects is $n \times (n - 1) \times (n - 2) \times . . . \times 2 \times 1$ divided by $r \times (r - 1) \times (r - 2) \times . . . \times 2 \times 1$

Worked Example

By considering the number of options there are for each letter to go into each position, find how many different arrangements there are of the letters in the word REVISE.

NMiqFoMp_2-2-1-arrangements-we-solution-1

Factorials

What are factorials?

Factorials are a type of mathematical operation (just like +, -, ×, ÷)
The symbol for factorial is !
- So to take a factorial of any non-negative integer, $n$ , it will be written $n$ ! And pronounced ‘ $n$ factorial’
The factorial function for any integer, $n$ , is $n! = n \times (n - 1) \times (n - 2) \times . . . \times 2 \times 1$
- For example, 5 factorial is 5! = 5 × 4 × 3 × 2 × 1
The factorial of a negative number is not defined
- You cannot arrange a negative number of items
0! = 1
- There are no positive integers less than zero, so zero items can only be arranged once
Most normal calculators cannot handle numbers greater than about 70!, experiment with yours to see the greatest value of $x$ such that your calculator can handle $x!$

How are factorials and arrangements linked?

The number of arrangements of $n$ different objects is $n!$
- Where $n! = n \times (n - 1) \times (n - 2) \times . . . \times 2 \times 1$
The number of different arrangements of $n$ objects with one object repeated $r$ times and the others all different is $\frac{n!}{r!}$
The number of different arrangements of $n$ objects with one object repeated $r$ times, another object repeated $s$ times and the other objects all different is $\frac{n!}{r! s!}$

What are the key properties of using factorials?

Some important relationships to be aware of are:
- $n! = n \times (n - 1)!$
  - Therefore

$\frac{n!}{(n - 1)!} = n$

- $n! = n \times (n - 1) \times (n - 2)!$
  - Therefore

$\frac{n!}{(n - 2)!} = n \times (n - 1)$

Expressions with factorials in can be simplified by considering which values cancel out in the fraction
- Dividing a large factorial by a smaller one allows many values to cancel out

$\frac{8!}{5!} = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} = 8 \times 7 \times 6$

Worked Example

(i)

Show, by writing 8! and 5! in their full form and cancelling, that

$\frac{8!}{5!} = 8 \times 7 \times 6$

(ii)

Hence, simplify

\frac{n!}{(n - 3)!}

(iii)

The letters A, B, B, B, B, B, C and D are arranged in a row. How many different ways are there to arrange the 8 letters in a row?

2-2-1-factorials-we-solution-2-

Exam Tip

Arrangements and factorials are tightly interlinked with permutations and combinations. Make sure you fully understand the concepts in this revision note as they will be fundamental to answering perms and combs exam questions.

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CIE A Level Maths: Probability & Statistics 1

Revision Notes

2.2.1 Arrangements & Factorials

Arrangements

How many ways can n different objects be arranged?

What happens if the objects are not all different?

Worked Example

Factorials

What are factorials?

How are factorials and arrangements linked?

What are the key properties of using factorials?

Worked Example

Exam Tip

Author: Amber

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