Our advanced Permutation and Combination Calculator helps you solve complex combinatorics problems instantly. Whether you need to find the number of possible arrangements (permutations) or selections (combinations), this tool provides accurate results with detailed formulas for students and professionals alike.
Calculation Tool
P(n, r) = n! / (n – r)! = 5! / (5-3)! = 60
Permutation & Combination Methods
Different methods are used depending on whether order matters and if repetition is allowed. Here’s a breakdown:
Permutations (nPr)
Calculates the number of ways to arrange ‘r’ items from a set of ‘n’ items where the order is important.
- Order matters: (A, B, C) is different from (C, B, A).
- No repetition: Each item can be used only once.
- Formula: n! / (n – r)!
Combinations (nCr)
Calculates the number of ways to choose ‘r’ items from ‘n’ items where order is not important.
- Order doesn’t matter: {A, B, C} is the same as {C, B, A}.
- No repetition: Each item can be selected only once.
- Formula: n! / [r! × (n – r)!]
Variations with Repetition
Calculates the number of arrangements where order matters and items can be repeated.
- Order matters: (A, A, B) is different from (A, B, A).
- Repetition allowed: Items can be chosen multiple times.
- Formula: nr
The Relationship Between Permutations and Combinations
A simple way to understand the connection is to see that a permutation is a combination that is subsequently ordered. The number of ways to order ‘r’ items is r! (r factorial).
Therefore, the number of permutations is the number of combinations multiplied by the number of ways to arrange the selected items:
P(n, r) = C(n, r) × r!
This highlights the core difference: combinations are about choosing a group, and permutations are about arranging that group. Because permutations involve an extra step (ordering), the number of permutations is always greater than or equal to the number of combinations for the same ‘n’ and ‘r’.
Special Cases in Permutations
Beyond the standard formulas, combinatorics includes special cases for specific scenarios.
Permutations with Identical Items
This calculates arrangements for a set where some items are identical (a multiset).
- Scenario: Arranging the letters in the word “SUCCESS”.
- Formula: n! / (n1! n2! … nk!)
- Explanation: Here, n is 7. We divide by the factorials of counts of each repeated letter (S:3, C:2).
- Calculation: 7! / (3! × 2!) = 420 ways.
Circular Permutations
This is used when arranging items in a circle, where rotations are not considered distinct.
- Scenario: Seating 6 people around a circular table.
- Formula: (n – 1)!
- Explanation: We fix one person’s position and arrange the remaining (n-1) people relative to them.
- Calculation: (6 – 1)! = 5! = 120 ways.
Practical Examples
Combinatorics are used to solve problems in many fields. Here are a few examples:
Team Selection
How many ways can a coach choose a starting lineup of 5 players from a team of 12?
Lock Combinations
How many 3-digit codes are possible for a lock using digits 0-9 if repetition is allowed?
Event Scheduling
In what order can 4 different speakers be scheduled to present at a conference?
Frequently Asked Questions
The key difference is order. Permutations are arrangements where order matters (e.g., ABC and CBA are different). Combinations are selections where order does not matter (e.g., {A,B,C} and {C,B,A} are the same group).
Use permutations for ordered lists, like arranging books on a shelf, race results, or assigning specific roles. Use combinations for unordered groups, like choosing a committee, picking lottery numbers, or selecting toppings for a pizza.
It depends. If you are choosing ‘r’ items from ‘n’ with repetition allowed, use nr (our “Variation” option). If arranging a set with identical items (like letters in “APPLE”), use the multiset formula: n! / (n1! n2! …), where n1 is the count of the first repeated item, etc.
A circular permutation is an arrangement of items in a circle. Since there’s no start or end point, rotations are considered the same arrangement. The formula is (n – 1)!, where ‘n’ is the number of items. For example, arranging 4 people around a table is (4-1)! = 3! = 6 ways.
0! is defined to be 1. This is a mathematical convention that allows formulas like the permutation formula P(n,n) = n! / (n-n)! to work correctly. It represents one way to arrange nothing (the arrangement of an empty set).
Yes. When n = r, you are calculating the number of ways to arrange all items in a set. The formula P(n,n) simplifies to n! / (n-n)! = n! / 0! = n!, which is the total number of unique ways to order the entire set.
Yes. This is often called a “multiset coefficient” or solved with the “stars and bars” method. The formula is C(n+r-1, r). It calculates ways to choose ‘r’ items from ‘n’ options where order doesn’t matter and repetition is allowed, like picking 3 scoops of ice cream from 5 flavors.
In machine learning, permutation tests are used to determine feature importance. By shuffling the values of a single feature and measuring how much the model’s accuracy drops, data scientists can gauge how critical that feature is for making correct predictions.
Factorial (n!) is the product of all positive integers up to n (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120). It’s a cornerstone of permutation and combination formulas because it represents the total number of ways to arrange n distinct items.
They are fundamental. Probability is often calculated as (Number of Favorable Outcomes) / (Total Number of Possible Outcomes). Permutations and combinations are used to find both the numerator and the denominator in this equation. For example, combinations can tell you the total possible poker hands, and the number of ways to get a specific hand (like a full house).