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 A-₁?and A₂?we have
    • A-_1?A₂?and A_2?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
    • A-_1?A_2?C, A-_2?A-_1?C, A-_1?---C A-₂, , A-_2?C A₁, C A-_1?---A-₂, C A-_2?A-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

• • 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
    • A-_1??A₂?A-₃?, A_1??A_3??A-_2??, A-_2??A₁?A-₃?, A_2?A_3?A-₁?, A-_3?A₁?A-₂? ,? A_3?A_2?A-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

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?