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

---

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

---

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

---

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

---

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)

---

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

---

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

---

Related Resources

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

Frequently Asked Questions

What is the difference between combination and permutation?

In permutations, ORDER MATTERS (arranging items in sequence). In combinations, ORDER DOESN’T MATTER (selecting items from a group). Permutations of ABC: ABC, ACB, BAC, BCA, CAB, CBA (6 ways). Combinations of ABC: just ABC (1 way to select all three).

How do I calculate nCr?

nCr (n choose r) = n! / (r! × (n-r)!). For example, 5C3 = 5! / (3! × 2!) = 120 / (6 × 2) = 10. This gives the number of ways to choose r items from n items when order doesn’t matter.

How do I calculate nPr?

nPr = n! / (n-r)!. For example, 5P3 = 5! / 2! = 120 / 2 = 60. This gives the number of ways to arrange r items from n items when order matters.

When should I use permutations vs combinations?

Ask: “Does the order of selection matter?” If rearranging items creates a different outcome (like rankings, passwords, or seat assignments), use permutations. If not (like lottery numbers, committees, or card hands), use combinations.

What does “with repetition” mean?

Repetition allows items to be used more than once. Example: a 4-digit PIN can reuse digits (1111 is valid), so use permutation with repetition: 10⁴ = 10,000 possibilities. Without repetition, each digit can only appear once: P(10,4) = 5,040.

How do I calculate combinations with large numbers?

Use the formula step by step, canceling common factors before multiplying. For C(52,5): 52×51×50×49×48 / (5×4×3×2×1). Cancel: (52×51×50×49×48) / 120 = 2,598,960. Or use this calculator for instant results.