Permutation and Combination
Permutations and combinations are counting methods used to determine how many different ways items can be arranged or selected. The main difference is whether the order of the selected items matters. Permutations are used when order matters, while combinations are used when order does not matter.
What Is a Permutation?
A permutation is an arrangement of items where the order is important. For example, choosing the first, second, and third positions in a competition creates different outcomes depending on who finishes in each position.
The formula for selecting r items from n items is:
nPr = n! Γ· (n β r)!
Here, n is the total number of available items, r is the number of items being arranged, and ! represents the factorial operation.
Permutation Example
Suppose you have 5 students and want to choose 3 of them for first, second, and third place.
5P3 = 5! Γ· (5 β 3)!
5P3 = 5! Γ· 2! = 60
There are 60 possible arrangements because changing the order creates a different result.
What Is a Combination?
A combination is a selection where the order does not matter. For example, choosing 3 students for a team gives the same group regardless of which student is selected first.
The formula is:
nCr = n! Γ· [r!(n β r)!]
As with permutations, n represents the total number of items and r represents the number being selected.
Combination Example
Suppose you have 5 students and need to select 3 for a team.
5C3 = 5! Γ· [3!(5 β 3)!]
5C3 = 10
There are 10 possible groups. The order of the students within the team does not create a new combination.
Permutation vs Combination
| Feature | Permutation | Combination |
|---|---|---|
| Order matters | Yes | No |
| Used for | Arrangements and rankings | Groups and selections |
| Formula | n! Γ· (n β r)! | n! Γ· [r!(n β r)!] |
When Should You Use Each One?
Use a permutation when changing the order creates a different outcome. Examples include passwords, race positions, seat arrangements, and ranking participants.
Use a combination when the selected group stays the same regardless of order. Examples include selecting a committee, choosing lottery numbers, or forming a team.
Understanding Factorials
Factorials are used in both permutation and combination formulas. The factorial of a positive whole number is the product of that number and all positive integers below it.
For example:
5! = 5 Γ 4 Γ 3 Γ 2 Γ 1 = 120
By definition, 0! = 1. This rule is important when working with combinations and other counting formulas.
Common Mistakes to Avoid
The most common mistake is choosing the wrong method. Ask yourself whether changing the order of the selected items creates a different result. If it does, use a permutation. If it does not, use a combination.
Also make sure that r is not greater than n, because you cannot select or arrange more items than are available.
Quick Summary
Permutations count arrangements where order matters, while combinations count selections where order does not matter. The correct method depends on the type of problem you are solving.