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Combinations Calculator

Calculate combinations (nCr) and permutations (nPr) instantly.

Order matters or notnCr and nPrExact whole-number results

n is the total number of available items (whole number, 0 to 300).

r is how many items you choose (whole number, 0 to n; up to 300 with replacement).

Exact BigInt arithmetic · copies full digits

10C3 = 120. 10P3 = 720.

Combination Result

10C3 = 120

Permutation Result

10P3 = 720

n!

3,628,800

r!

6

(n − r)!

5,040

10C3 = 120: the number of unordered ways to choose 3 items from 10.
10P3 = 720: the number of ordered arrangements, because order matters.
The permutation result is usually larger because it counts different orders separately.
Combinations are commonly used for teams, groups, or selections.
Permutations are commonly used for rankings, seating, passwords, or ordered choices.
Without replacement means an item cannot be reused.

Formulas

Combination

nCr = n! ÷ (r! × (n − r)!)

Permutation

nPr = n! ÷ (n − r)!

Factorial

n! = n × (n − 1) × ... × 1

Combination With Replacement

(n + r − 1)Cr

Permutation With Replacement

n^r

Variable Explanations

n

Total number of available items.

r

Number of items selected or arranged.

!

Factorial operator.

nCr

Combinations where order does not matter.

nPr

Permutations where order matters.

n − r

Items not selected.

With replacement

Items can repeat.

Without replacement

Items cannot repeat.

Combinations vs Permutations

Combinations

Combinations count selections where order does not matter.

Choosing Alice and Bob is the same as choosing Bob and Alice.

Permutations

Permutations count arrangements where order matters.

First Alice then Bob is different from first Bob then Alice.

Worked Examples

5C2

Formula: 5! ÷ (2! × 3!)

Answer: 10

5P2

Formula: 5! ÷ 3!

Answer: 20

Choosing 3 winners from 10 people

Formula: 10C3

Answer: 120

Arranging 3 podium places from 10 runners

Formula: 10P3

Answer: 720

Committee selection

Formula: Order does not matter

Answer: Combination

Password example

Formula: Order matters, characters can repeat

Answer: Permutation with replacement (n^r)

Choosing a 5-card hand from a standard 52-card deck

Formula: C(52, 5) = 52! ÷ (5! × 47!)

Answer: 2,598,960

Cards are dealt without replacement and order does not matter: the same five cards in any order are one hand.

Choosing 3 items from 10 types, repetition allowed

Formula: C(10 + 3 − 1, 3) = C(12, 3)

Answer: 220

With replacement the same type can be chosen more than once, and order still does not matter.

Factorials and Counting Principles

Factorials multiply descending positive integers.
Factorials grow very quickly.
Combinations divide out duplicate orderings.
Permutations keep orderings.
The counting principle multiplies choices across steps.
For 3 ordered choices from 5: 5 × 4 × 3 = 60.

Common Use Cases

Lottery odds
Card hands
Team selection
Committee selection
Seating arrangements
Race rankings
Passwords and PINs
Tournament brackets
Product bundles

Common Mistakes

Using permutations when order does not matter.
Using combinations when order matters.
Entering r greater than n without replacement.
Forgetting factorials grow quickly.
Confusing replacement and non-replacement.
Double-counting arrangements.
Assuming nCr and nPr are interchangeable.
Forgetting that 0! = 1.

Frequently Asked Questions

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