A permutation is an arrangement of objects in a specific order. In permutations, the order matters.
For example, if you have the letters A, B, and C, the possible permutations are:
- ABC
- ACB
- BAC
- BCA
- CAB
- CBA
There are 6 different arrangements because changing the order creates a new permutation.
For the general case of choosing and arranging r objects from n different objects:
The formula is:
{}^nP_r = \frac{n!}{(n-r)!}
where:
- n = total number of objects
- r = number of objects to arrange
- ! (factorial) means multiplying a number by all positive integers below it (e.g., 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120).
Example
How many ways can you arrange 2 letters from A, B, C?
The arrangements are:
- AB
- AC
- BA
- BC
- CA
- CB
There are 6 permutations.
Using the formula:
{}^3P_2 = \frac{3!}{(3-2)!} = \frac{6}{1} = 6
Permutation vs. Combination
- Permutation: Order matters. Example: AB and BA are different.
- Combination: Order does not matter. Example: AB and BA are the same.
A simple way to remember it is:
Permutation = Position matters; Combination = Choice matters.
