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Permutation Combination Calculator | Calculate nPr And nCr

Calculate permutations (nPr) and combinations (nCr) with the full factorial working shown. Perfect for probability problems, lottery odds, scheduling arrangements and counting questions — enter n and r and get both values, plus the reasoning behind each.

Last updated: July 2026 · Free · No sign-up required

Results

Enter values above and click Calculate to see your result instantly.

Quick reference

Permutations vs combinations:

QuestionUse
Does order matter?Permutation (nPr)
Order does not matter?Combination (nCr)
Example: race finishesPermutation
Example: lottery picksCombination

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For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.

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Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.

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Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.

Interpretation guide

Sample values (n = 5):

rnPr (5)nCr (5)
155
22010
36010
51201

lightbulb Worked example

Let's say you are using the Permutation and Combination Calculator. Enter n and r and this calculator computes both the number of permutations (nPr) and combinations (nCr), with the full factorial working shown. It explains the difference — order matters for Enter the values that match your situation into the input fields and press calculate — using realistic numbers makes the result directly useful for you.

Result: The calculator instantly applies the formula nPr = n! / (n - r)!; nCr = n! / (r! (n - r)!) and returns the result with appropriate precision.

What this means: Read the result in the context of what you are measuring. The step-by-step breakdown lets you confirm the math and understand which input most affects the outcome.

Formula & methodology

Formula: nPr = n! / (n - r)!; nCr = n! / (r! (n - r)!)

The Permutation and Combination Calculator is built on a well-established calculation method. It uses the formula nPr = n! / (n - r)!; nCr = n! / (r! (n - r)!) to turn your inputs into a reliable result. Enter n and r and this calculator computes both the number of permutations (nPr) and combinations (nCr), with the full factorial working shown. It explains the difference — order matters for permutations, not for The steps are shown on the page so you can follow the reasoning from input to output.

Permutations count ordered arrangements; combinations count unordered selections. The calculator evaluates nPr = n!/(n-r)! when order matters (like ranking finishers) and nCr = n!/(r!(n-r)!) when it doesn't (like choosing a committee). Because every combination corresponds to r! permutations, nPr is always r! times larger than nCr. These counts power probability, lottery odds, and combinatorics problems.

Authoritative source: Wolfram MathWorld

Frequently asked questions

What's the difference between a permutation and a combination?
The Permutation and Combination Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Enter n and r and this calculator computes both the number of permutations (nPr) and combinations (nCr), with the full factorial working shown. It explains the difference — order matters for permutations, not for combinations — and includes the repetition There is no limit on usage, and it works on any device.
How do I calculate nCr?
Enter the required values into the input fields and press the calculate button. The tool applies the formula nPr = n! / (n - r)!; nCr = n! / (r! (n - r)!) and returns the result with a short explanation of each step, so you can check the working yourself.
What is a factorial?
A factorial, written n!, is the product of every positive integer from 1 to n: 5! = 5×4×3×2×1 = 120. By convention, 0! = 1. Factorials count arrangements and grow extremely fast — 10! is 3,628,800 and 20! is about 2.4 quintillion. They are the building blocks of permutation and combination formulas, which the calculator expands step by step.
How are lottery odds calculated?
Lottery odds are a combination problem: the number of ways to choose r winning numbers from n total is nCr = n! ÷ (r! × (n−r)!). For a 6-of-49 lottery, that is 49C6 = 13,983,816 combinations, so your chance of matching all six is 1 in 13,983,816. The calculator computes the combination and converts it to the odds ratio directly.
Why is nPr larger than nCr?
Because permutations count arrangements while combinations count selections — for the same n and r, every combination can be arranged in r! different orders, so nPr = nCr × r!. Choosing 3 items from 5 gives 10 combinations but 60 permutations, since each set of 3 can be ordered 6 ways. The calculator shows both values together so the relationship is visible.
What's the difference between nCr and nPr?
nPr counts ordered arrangements, nCr counts unordered selections. Picking a president and vice-president from 10 people is P(10,2) = 90; picking a 2-person committee is C(10,2) = 45 — exactly half, because each pair was counted twice when order mattered. Divide permutations by r! and you get combinations.
How do I calculate combinations by hand?
C(n,r) = n! ÷ (r!(n−r)!), but cancel before multiplying: C(8,3) = (8 × 7 × 6) ÷ (3 × 2 × 1) = 56. Take r factors descending from n on top, r! on the bottom. Also remember the symmetry C(n,r) = C(n,n−r) — C(50,47) is just C(50,3) = 19,600.
What does the Permutation and Combination Calculator do?
Enter n and r and this calculator computes both the number of permutations (nPr) and combinations (nCr), with the full factorial working shown. It explains the difference — order matters for permutations, not for combinations — and includes the repetition. The calculator takes your inputs, applies the standard calculation, and returns a clear result so you can make an informed decision without doing the math by hand.
What formula does the Permutation and Combination Calculator use?
It uses nPr = n! / (n − r)! for permutations, where order matters, and nCr = n! / (r! × (n − r)!) for combinations, where order does not. The full factorial working is shown so you can follow each step. For example, 5C2 = 120 / (2 × 6) = 10 ways to choose 2 items from 5.
Is the Permutation and Combination Calculator free and private?
Yes — the Permutation and Combination Calculator. There is no sign-up, no paywall, no trial, and no limit on how many calculations you can run. CalcaTools covers its costs with non-intrusive display advertising, so the calculator itself never asks for payment or restricts any feature.
How accurate is the Permutation and Combination Calculator?
The Permutation and Combination Calculator applies the standard formula: nPr = n! / (n - r)!; nCr = n! / (r! (n - r)!). It is tested against published worked examples before launch and uses native double-precision arithmetic, so results are accurate to typical precision for the values you enter — with no intermediate rounding that could distort the answer.

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