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CIE A Level Maths: Probability & Statistics 1复习笔记2.2.1 Arrangements & Factorials

Category: A-level课程, 教材笔记, 福利干货 Date: 2022年8月18日上午11:13

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

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

- 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

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 !

How are factorials and arrangements linked?

What are the key properties of using factorials?

Worked Example

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.

转载自savemyexams