Arrangements & Factorials (Cambridge (CIE) A Level Maths): Revision Note

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 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

      • ABC, ACB, BAC, BCA, CAB and CBA

    • 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  

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

  • A­₁ A₂ and A₂ A­­_­1

  • 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

  • A­₁ A₂ C, A­₂ A­₁ C, A­₁ ­­­C A­₂, , A­₂ C A₁, C A­₁ ­­­A­₂, C A­₂ A­₁

  • 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  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

  • A­₁  A₂ A­₃ , A₁  A₃  A­₂  , A­₂  A₁ A­₃ , A₂ A₃ A­₁ , A­₃ A₁ A­₂  ,  A₃ A₂ A­₁

  • 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  = 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  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  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   divided by  

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, , it will be written !  And pronounced ‘ factorial’

• The factorial function for any integer, , is 

  • 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  such that your calculator can handle 

How are factorials and arrangements linked?

• The number of arrangements of  different objects is 

  • Where 

• The number of different arrangements of objects with one object repeated times and the others all different is 

• The number of different arrangements of objects with one object repeated times, another object repeated times and the other objects all different is 

What are the key properties of using factorials?

• Some important relationships to be aware of are:

  • 

    • Therefore

• 

  • Therefore

• 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