When should I use combinations instead of permutations?

Use combinations when you only care about which items are selected, not the order. Typical examples include committees, lottery numbers, or subsets. Use permutations when arrangement or sequence matters.

What do nCr and nPr mean in this calculator?

nCr is the number of combinations, given by n! divided by r!(n − r)!, and counts unordered selections of r objects from n. nPr is the number of permutations, given by n! divided by (n − r)!, and counts ordered arrangements.

How do combinations and permutations change with repetition allowed?

With repetition allowed, permutations are counted by n^r and combinations by (n + r − 1)! divided by r!(n − 1)! (the stars and bars formula).

Combination and Permutation Calculator (nCr & nPr)

Compute combinations and permutations for any valid n and r, with or without repetition. Review exact integer results, digit counts, and optional factorial breakdowns tailored to counting problems.

Input parameters

Total items (n)

n is the total number of distinct objects. Recommended range: 0–500. For factorial steps keep n ≤ 50.

Chosen/arranged items (r)

r is how many items you select or arrange. Without repetition, r cannot exceed n.

Type of counting

Both combinations (nCr) and permutations (nPr) Combinations only (nCr) Permutations only (nPr)

Repetition

No repetition allowed Repetition allowed

Permutations with repetition use n^r; combinations with repetition use C(n + r − 1, r).

Summary

Enter inputs and press Calculate.

Combinations —

Permutations —

Digits (largest) —

Scientific notation —

Enter values for n and r, choose the options above, then click Calculate to see combinations and permutations.

How to Use This Calculator

Enter the total number of distinct objects (n), how many you want to select or arrange (r), the counting mode, and whether repetition is allowed. Click Calculate to compute the exact integers and view metadata such as digit counts and the factorial breakdown.

Methodology

The calculator uses big integer math to evaluate factorials, binomial coefficients, and exponentiation so the results remain exact. The factorial expansion is provided for n and r ≤ 50 to keep the walkthrough readable. No floating-point rounding occurs, and intermediate values are formatted consistently for easy verification.

Definitions

Permutation: Ordered arrangement of r objects chosen from n, so swapping order produces a new permutation.
Combination: Unordered selection of r objects from n; the same elements count once regardless of order.

Typical use cases

Combinations: Lottery tickets, committees, subsets of features, portfolios where order is irrelevant.
Permutations: Seating arrangements, order-sensitive codes, rankings, or priority-sensitive assignments.

Worked example

Suppose n = 10 books and r = 3 are chosen.

Permutations: \( {}_{10}P_3 = \frac{10!}{(10-3)!} = \frac{10!}{7!} = 10 \cdot 9 \cdot 8 = 720 \)
There are 720 ordered arrangements of three books.
Combinations: \( {}_{10}C_3 = \frac{10!}{3! \cdot 7!} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120 \)
There are 120 groups of three books ignoring order.

FAQ – Combinations & permutations

What is the difference between a permutation and a combination?

A permutation counts ordered arrangements; swapping order yields a different permutation. A combination counts unordered selections, so permutations that differ only by order collapse into the same combination.

When should I use permutations instead of combinations?

Use permutations when order matters: rankings, seats, sequences, or codes where position is important. Use combinations when you only care which items are included—committee selection, card hands, or subsets.

How large can n and r be?

The calculator accepts values up to n = 500 and r = 500, using big integers for exactness. Factorial walkthroughs remain readable for n, r ≤ 50; larger values still compute precise totals but omit step-by-step terms.

Why are combinations smaller than permutations for the same n and r?

Each combination of r objects can be arranged in r! different ways. Permutations count each arrangement separately, while combinations collapse those r! permutations into one outcome. Hence \( {}_nP_r = {}_nC_r \cdot r! \).

Full original guide (expanded)

Related Core Math & Algebra tools

Permutation and Combination (theory overview) Combination Calculator (nCr) Permutation Calculator (nPr) Factorial Calculator (n!) Binomial Distribution Calculator Lottery Odds Calculator Probability Calculator Standard Deviation Calculator

Quick reminders

Permutation = order matters.
Combination = order does not matter.
\( {}_nP_r = {}_nC_r \cdot r! \).
With repetition: permutations \( n^r \ ), combinations \( \binom{n + r - 1}{r} \ ).

Formulas

Permutations (no repetition)

\( {}_nP_r = \frac{n!}{(n-r)!} \)

Combinations (no repetition)

\( {}_nC_r = \frac{n!}{r!(n-r)!} \)

Permutations with repetition

\( P_{\text{rep}}(n,r) = n^r \)

Combinations with repetition

\( C_{\text{rep}}(n,r) = \binom{n + r - 1}{r} = \frac{(n + r - 1)!}{r!(n - 1)!} \)

Worked example

\( {}_{10}P_3 = \frac{10!}{7!} = 10 \cdot 9 \cdot 8 = 720 \)
\( {}_{10}C_3 = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120 \)

Citations

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures

FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

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