Combination & Permutation Calculator Online

What are Combinations and Permutations?

Combinations and permutations are fundamental concepts in combinatorics used to calculate the number of ways to select or arrange objects from a set. Although often mentioned together, they have a fundamental difference: permutations consider order, while combinations do not consider order. Permutation is the arrangement of objects in a specific order. The permutation formula P(n,r) = n! / (n-r)! calculates how many ways to arrange r objects from n available objects while paying attention to position/order. Example: Arranging 3 out of 5 books on a shelf - the order A-B-C is different from B-A-C. Combination is the selection of objects without considering order. The combination formula C(n,r) = n! / (r!(n-r)!) calculates how many ways to choose r objects from n objects regardless of order. Example: Choosing 3 out of 5 committee members - the team A,B,C is the same as B,A,C. The KalkuLab Combinations & Permutations Calculator helps students and professionals quickly calculate these values with automatic factorial computation and detailed step-by-step solutions.

Combinations and Permutations Formulas

Permutation: P(n,r) = n! / (n-r)! | Combination: C(n,r) = n! / (r!(n-r)!)Formula: P(n,n) = n! (Permutation of n objects) | C(n,0) = C(n,n) = 1

Variables:

nTotal Objects
Total number of objects available in the set(e.g.: If from 10 books, n=10)
💡 Population, total choices, data set
rObjects Selected
Number of objects to be chosen or arranged(e.g.: Choosing 3 from 10, r=3)
💡 Sample, committee, selected positions
n!Factorial
Product of all positive integers from 1 to n(e.g.: 5! = 5×4×3×2×1 = 120)
💡 Basis of combinatorics calculations
C(n,r)Combination
Number of ways to choose r objects from n without order(e.g.: C(5,3) = 10 ways)
💡 Team selection, lottery, committee
P(n,r)Permutation
Number of ways to arrange r objects from n with order(e.g.: P(5,3) = 60 ways)
💡 Word arrangement, positions, passwords

Steps for Solving Combinations/Permutations

First determine whether order matters (permutation) or not (combination). Calculate the required factorial values, then substitute into the appropriate formula.

1Determine n (total objects) and r (objects to select)
2Determine whether order matters (P) or not (C)
3Calculate factorials: n!, r!, (n-r)!
4Substitute into formula: P(n,r) = n!/(n-r)! or C(n,r) = n!/(r!(n-r)!)

Categories:

Combination (C)Order does NOT matter

Permutation (P)Order DOES matter

Factorial (n!)n × (n-1) × ... × 1

Special CombinationsC(n,0)=C(n,n)=1

How to Use the Combinations & Permutations Calculator on KalkuLab

Using the KalkuLab combinations and permutations calculator is very easy. Follow these steps:

1
Select Calculation Type
Choose whether you want to calculate Combination (C) or Permutation (P) from the available menu options.
2
Enter Value n (Total Objects)
Enter the total number of objects available (n). Make sure n is a non-negative integer and is large enough to select r objects.
3
Enter Value r (Objects to Select)
Enter the number of objects to be chosen or arranged (r). Make sure r ≤ n for valid calculations.
4
Press the Calculate Button
Click the calculate button to process the computation. The result will be displayed along with detailed step-by-step solutions.

💡 Tip:

•Make sure r ≤ n, since you cannot select more objects than are available
•Use the reset button to clear all values and start a new calculation
•Large factorial values (like 20!) can be very large, our calculator handles them
•Combinations always produce smaller or equal values compared to permutations for the same n and r

Examples

Example 1: Forming a Student Council Committee (Combination)

Problem:

A school wants to form a student council core committee of 5 students from 12 candidates. How many ways can the committee be formed if positions are not distinguished?

Solution:

1.Given: n = 12 (candidates), r = 5 (to select)
2.Use combination: C(12,5) = 12! / (5! × 7!)
3.C(12,5) = (12×11×10×9×8) / (5×4×3×2×1)
4.C(12,5) = 95040 / 120

Result:There are 792 ways to form the committee

There are 792 different ways to select 5 students from 12 candidates without distinguishing positions. The combination concept is very common in team or committee selection.

Example 2: Creating a Password (Permutation)

Problem:

A security system requires a 4-digit password using digits 0-9 without repetition. How many possible passwords can be created?

Solution:

1.Given: n = 10 (digits 0-9), r = 4 (password digits)
2.Use permutation: P(10,4) = 10! / (10-4)!
3.P(10,4) = 10! / 6! = 10×9×8×7

Result:There are 5,040 possible passwords

There are 5,040 different ways to arrange a 4-digit password without repetition. Permutations are very important in cryptography and system security.

Frequently Asked Questions

What is the fundamental difference between combination and permutation?

The main difference is order. Permutations consider order/position (A-B-C is different from B-A-C), while combinations do not consider order (A-B-C is considered the same as B-A-C). Use permutation if order matters, combination if order does not matter.

When should I use P(n,r) formula and when C(n,r)?

Use P(n,r) when arranging objects in a specific order (positions, passwords, word arrangements). Use C(n,r) when only selecting objects regardless of order (teams, menus, committees). Ask yourself: 'Is A-B the same as B-A?' If yes, use combination.

What is a factorial and how do you calculate it?

A factorial (n!) is the product of all positive integers from 1 to n. Example: 5! = 5×4×3×2×1 = 120. By definition, 0! = 1. Factorials grow very quickly: 10! = 3,628,800, 20! ≈ 2.4 × 10¹⁸.

Can r exceed n in combinations/permutations?

No. The value r (objects selected) cannot exceed n (total available objects). If r > n, then C(n,r) = P(n,r) = 0 because it is impossible to select more objects than are available.

Why are combination results always smaller than or equal to permutation results?

Because permutations account for order, so each combination (without order) has r! corresponding permutations. Mathematically: P(n,r) = C(n,r) × r!. Thus P(n,r) ≥ C(n,r) for all n and r.

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