Permutation and Combination Calculator
Calculate permutations and combinations quickly and accurately using n and r.
What Are Permutations and Combinations?
Permutations and combinations are methods used to count the number of ways objects can be selected from a set. The main difference is whether the order of selection matters.
Permutations are arrangements where the order of selected objects matters.
Combinations are selections where the order of the selected objects does not matter.
Permutations
A permutation is an arrangement of objects in which the order of the selected objects matters. For example, arranging three objects in different orders produces different permutations.
nPr = n! / (n − r)!
If n = 5 and r = 2:
5P2 = 5! / (5 − 2)!
5P2 = 5! / 3! = 20
Permutations are useful when the order of the selected objects or positions is important.
Combinations
A combination is a selection of objects in which the order of the selected objects does not matter. Different arrangements of the same selected objects are considered the same combination.
nCr = n! / [r! × (n − r)!]
If n = 5 and r = 2:
5C2 = 5! / [2! × (5 − 2)!]
5C2 = 5! / (2! × 3!) = 10
Combinations are useful when the order of the selected objects is not important.
Permutations Without Replacement
In permutations without replacement, an item cannot be selected more than once. The number of possible arrangements depends on both the total number of items and the number of items selected.
nPr = n! / (n − r)!
If there are 6 items and 3 are selected:
6P3 = 6! / (6 − 3)!
6P3 = 6! / 3! = 120
Combinations Without Replacement
In combinations without replacement, an item cannot be selected more than once. Since the order of selection does not matter, different arrangements of the same selected items are counted as one combination.
nCr = n! / [r! × (n − r)!]
If there are 6 items and 3 are selected:
6C3 = 6! / [3! × (6 − 3)!]
6C3 = 6! / (3! × 3!) = 20
How to Use the Permutation and Combination Calculator
Frequently Asked Questions
What is a permutation?
A permutation is an arrangement of objects where the order of the selected objects matters.
What is a combination?
A combination is a selection of objects where the order of the selected objects does not matter.
What is the difference between permutations and combinations?
The main difference is that order matters in permutations, while order does not matter in combinations.
What do n and r represent?
n represents the total number of objects in the set, while r represents the number of objects selected from the set.
Can r be greater than n?
No. The number of selected objects cannot be greater than the total number of objects. The calculator will display an error if r is greater than n.
Can I enter zero for n or r?
Yes. Zero is accepted as a whole-number input when it satisfies the mathematical conditions of the calculation.
What does without replacement mean?
Without replacement means that an object cannot be selected more than once.
Disclaimer
This calculator is provided for general mathematical and educational purposes. While we aim to provide accurate results, SmartMyTools does not guarantee that the results are suitable for professional, academic, financial, engineering, or other specialized applications.