Assigning officer roles
When different positions matter, changing the order creates a new outcome, so you use permutations.
P(5, 3) = 5! / (5-3)!
Result: 60 ways to assign 3 distinct roles from 5 people
Permutation and Combination Calculator
Use this calculator to compare permutations and combinations for the same values of n and r. It is useful for probability, counting problems, team selection, and exam prep.
What is a Permutation?
A permutation is an arrangement of objects where order matters. The formula for permutation is: \[P(n,r) = \frac{n!}{(n-r)!}\] where n is the total number of items and r is the number of items being arranged. For example, the number of ways to arrange 3 books out of 5 books is P(5,3) = 60.
What is a Combination?
A combination is a selection of objects where order doesn't matter. The formula for combination is: \[C(n,r) = \frac{n!}{r!(n-r)!}\] where n is the total number of items and r is the number of items being selected. For example, the number of ways to select 3 books out of 5 books is C(5,3) = 10.
Examples and Applications
Choosing a committee
When you only care which people are selected and not their order, combinations are the correct count.
C(5, 3) = 5! / (3!(5-3)!)
Result: 10 ways to choose 3 people from 5
Comparing the same n and r
For the same values, permutations are always greater than or equal to combinations because they count more outcomes.
P(8, 2) vs C(8, 2)
Result: P(8, 2) = 56 and C(8, 2) = 28
How to Use This Calculator
- Enter the total number of available items as n.
- Enter how many items you want to arrange or select as r.
- Click calculate to see both the permutation and combination results.
- Use the two results to understand whether order changes the count in your problem.
If the scenario involves ranked positions, seat assignments, or ordered passwords, focus on permutations. If the scenario is simple selection without order, focus on combinations.
Permutation and Combination FAQ
What is the main difference between permutation and combination?
Permutation counts arrangements where order matters, while combination counts selections where order does not matter.
Why must r be less than or equal to n?
You cannot choose or arrange more items than the total number available, so r cannot be larger than n.
When should I use permutations in probability?
Use permutations when the order of outcomes changes the event, such as podium finishes, lock codes without repetition, or ordered task assignments.
Why is the combination result smaller than the permutation result?
Combinations group together outcomes that differ only by order, so they count fewer unique cases than permutations.