Combinations and Permutations Calculator

Calculation Type

Combination

Order doesn't matter

Permutation

Order matters

Combination (Repetition)

With replacement

Permutation (Repetition)

With replacement

n - Total items

r - Items to choose

CalculateClear

Quick Reference

Type	Formula	Example
Permutation	n!/(n-r)!	Arranging 3 from 5: P(5,3)=60
Combination	n!/(r!(n-r)!)	Choosing 3 from 5: C(5,3)=10
Perm. w/ Rep.	n^r	3-digit code (0-9): 10³=1000
Comb. w/ Rep.	C(n+r-1,r)	5 scoops, 3 flavors: C(7,5)=21

Common Combinations C(n,r)

C(5,2)

10

C(6,3)

20

C(10,5)

252

C(52,5)

2,598,960

C(49,6)

13,983,816

C(20,10)

184,756

How to Use the Combinations & Permutations Calculator

This calculator helps you solve counting problems by computing combinations and permutations with or without repetition.

Quick Reference

Type	Order Matters?	Repetition?	Formula	Example
Permutation	Yes	No	n!/(n-r)!	Arranging 3 books on a shelf
Combination	No	No	n!/(r!(n-r)!)	Choosing a committee of 3
Perm. w/ Rep.	Yes	Yes	n^r	4-digit PIN codes
Comb. w/ Rep.	No	Yes	(n+r-1)!/(r!(n-1)!)	Selecting scoops of ice cream

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Combinations (Order Doesn’t Matter)

Use combinations when you’re selecting items and the order doesn’t matter.

Formula: C(n,r) = n! / (r! × (n-r)!)

Also written as: ₙCᵣ, (n choose r), or $\binom{n}{r}$

Examples

• Choosing 5 cards from a deck: C(52,5) = 2,598,960
• Selecting 3 toppings from 8: C(8,3) = 56
• Lottery numbers (6 from 49): C(49,6) = 13,983,816

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Permutations (Order Matters)

Use permutations when you’re arranging items and the order matters.

Formula: P(n,r) = n! / (n-r)!

Also written as: ₙPᵣ or P(n,r)

Examples

• First, second, third place from 10 runners: P(10,3) = 720
• Arranging 4 books on a shelf: P(4,4) = 24
• Assigning 3 different prizes to 8 people: P(8,3) = 336

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With Repetition

Permutations with Repetition

When items can be reused in different positions.

Formula: n^r

Examples:

• 4-digit PIN (0-9): 10⁴ = 10,000
• License plate (3 letters): 26³ = 17,576
• Binary strings of length 8: 2⁸ = 256

Combinations with Repetition

When selecting items that can be chosen multiple times, and order doesn’t matter.

Formula: C(n+r-1, r)

Examples:

• 5 scoops from 3 flavors: C(7,5) = 21
• Distributing 10 identical items to 4 people: C(13,10) = 286

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Decision Guide

Ask yourself these questions:

1. Does order matter?

   • Yes → Use Permutation
   • No → Use Combination

2. Can items be repeated?

   • Yes → Use “with repetition” version
   • No → Use standard version

Scenario Examples

Scenario	Type	Calculation
Password with 6 characters (a-z, can repeat)	Perm. w/ Rep.	26⁶
Choosing a team of 5 from 12	Combination	C(12,5)
Ranking top 3 from 10 contestants	Permutation	P(10,3)
Ways to get 4 items from 6 categories	Comb. w/ Rep.	C(9,4)

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Common Values Table

Combinations C(n,r)

n\r	1	2	3	4	5	6
5	5	10	10	5	1	-
6	6	15	20	15	6	1
7	7	21	35	35	21	7
8	8	28	56	70	56	28
10	10	45	120	210	252	210
52	52	1,326	22,100	270,725	2,598,960	-

Permutations P(n,r)

n\r	1	2	3	4
5	5	20	60	120
6	6	30	120	360
7	7	42	210	840
10	10	90	720	5,040

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Properties and Identities

Symmetry

C(n,r) = C(n, n-r)

Example: C(10,3) = C(10,7) = 120

Pascal’s Triangle

C(n,r) = C(n-1,r-1) + C(n-1,r)

Sum of Row

C(n,0) + C(n,1) + … + C(n,n) = 2ⁿ

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Related Resources

• Counting Principles Lesson - Learn the fundamentals
• Factorial Calculator - Calculate n!
• Binomial Distribution - Uses combinations
• Probability Calculator - Calculate probabilities