We want to determine the similarities and differences between permutations and combinations. To do so, let's first recall their definitions.
Permutation
Combination
Definition
An arrangement of r objects at a time out of n objects in a specific order
A selection of r out of n objects without regard to order
Formula
_nP_r = n!/(n-r)!
_nC_r = n!/r!(n-r)!
We will find the similarities and differences one at a time.
Similarities
First, note that both permutations and combinations are used to find the number of possible outcomes in an experiment or possible ways to choose objects. It means that they are examples of counting methods.
Counting Methods
Used to find the number of possible outcomes in an experiment or possible ways to choose objects.
Also, both permutations and combinations are commonly used to find probabilities in different scenarios. These concepts will be taught in future lessons.
If there are m ways to make a first selection and n ways to make a second selection, then the total number of ways to make both selections is m* n.
Differences
A permutation is an arrangement in a specific order, while a combination is a selection without regard to order. Therefore, they differ in whether the order matters. Next, let's compare the formulas for the number of permutations and the number of combinations.
Number of Permutations
Number of Combinations
_nP_r=n!/(n-r)!
_nC_r=n!/r! (n-r)!
As we can see, the only difference between the formulas is the factor r! in the denominator of the formula for _nC_r.
Conclusion
Finally, we can summarize the similarities and differences between permutations and combinations.
Permutations Versus Combinations
Similarities
Differences
Examples of counting methods
Permutations are used when order is important, while combinations are used when selection — not order — is important.
Used to find probabilities in different scenarios
Different formulas for finding the number of possible ways to choose objects
