Arrangements & Factorials (CIE A Level Maths: Probability & Statistics 1)

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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 equals 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 begin mathsize 16px style n equals 3 end style  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 minus 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 cross times left parenthesis n minus 1 right parenthesis cross times left parenthesis n minus 2 right parenthesis cross times... cross times 2 cross 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
      • 1 A2 and A2 A­­­1
      • 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
      • 1 A2 C, A­2 1 C, A­1 ­­­C A­2, , A­2 C A1, C A­1 ­­­A­2, C A­2 1
      • 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 6 over 2 equals 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
      • A23 , AA, A­A13 , A2 A3 1 , A­3 A12  ,  A3 A2 1
      • 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 cross times 3 cross times 2 cross 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 cross times left parenthesis n minus 1 right parenthesis cross times left parenthesis n minus 2 right parenthesis cross times... cross times 2 cross times 1  divided by  r cross times open parentheses r minus 1 close parentheses cross times open parentheses r minus 2 close parentheses cross times... cross times 2 cross 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

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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 begin mathsize 16px style n factorial space equals space n space cross times left parenthesis n minus 1 right parenthesis cross times left parenthesis n minus 2 right parenthesis cross times... cross times 2 cross times 1 end style
    • 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 factorial

How are factorials and arrangements linked?

  • The number of arrangements of n different objects is n factorial
    • Where n factorial equals n cross times left parenthesis n minus 1 right parenthesis cross times left parenthesis n minus 2 right parenthesis cross times... cross times 2 cross times 1
  • The number of different arrangements of nobjects with one object repeated r times and the others all different is begin mathsize 16px style fraction numerator n factorial over denominator r factorial end fraction end style
  • The number of different arrangements of nobjects with one object repeated r times, another object repeated s spacetimes and the other objects all different is fraction numerator n factorial over denominator r factorial s factorial end fraction

What are the key properties of using factorials?

  • Some important relationships to be aware of are:
    • n factorial equals n cross times left parenthesis n minus 1 right parenthesis factorial
      • Therefore

begin mathsize 16px style fraction numerator n factorial over denominator left parenthesis n minus 1 right parenthesis factorial end fraction equals n end style

    • n factorial equals n cross times left parenthesis n minus 1 right parenthesis cross times left parenthesis n minus 2 right parenthesis factorial
      • Therefore

fraction numerator n factorial over denominator left parenthesis n minus 2 right parenthesis factorial end fraction equals n cross times left parenthesis n minus 1 right parenthesis

  • 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

fraction numerator 7 factorial over denominator 4 factorial end fraction equals fraction numerator 7 cross times 6 cross times 5 cross times 4 cross times 3 cross times 2 cross times 1 over denominator 4 cross times 3 cross times 2 cross times 1 end fraction equals 7 cross times 6 cross times 5

Worked example

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

fraction numerator 8 factorial over denominator 5 factorial end fraction equals 8 cross times 7 cross times 6

(ii)
Hence, simplify  fraction numerator n factorial over denominator left parenthesis n minus 3 right parenthesis factorial end fraction

 

(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-

Examiner 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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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.