Arrangements
How many ways can n different objects be arranged?
- When arranging different objects in a row, consider how many of the objects can go in the first position, how many can go in the second and so on
- For example, if there are two options for the first position and then there will only be one object left to go in the second position
- So a total of 2 × 1 = 2 possible arrangements
- To arrange the letters A and B we have
- AB and BA
- For example, if 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 a total of 3 × 2 × 1 = 6 possible arrangements
- To arrange the letters A, B and C we have
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- ABC, ACB, BAC, BCA, CAB and CBA
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- For example, if there are two options for the first position and then there will only be one object left to go in the second position
- 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
Worked example
By considering the number of options there are for each letter to go into each position, find how many distinct arrangements there are of the letters in the word MATHS.
There are 5 different letters in the word MATHS, so there are 5 letters for the first space, then there will be four for the second, three for the third and so on.
120 distinct arrangements