Permutation Combination Calculator | Calculate nPr And nCr

Quick reference

Permutations vs combinations:

Question	Use
Does order matter?	Permutation (nPr)
Order does not matter?	Combination (nCr)
Example: race finishes	Permutation
Example: lottery picks	Combination

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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):

r	nPr (5)	nCr (5)
1	5	5
2	20	10
3	60	10
5	120	1

lightbulb Worked example — leadership team vs committee from 12 volunteers

Twelve volunteers are available. First, choose a president, vice-president and secretary (order matters). Then, choose any 3-person committee (order doesn’t matter).

Result: Leadership team: P(12,3) = 12 × 11 × 10 = 1,320 possible ordered teams. Committee: C(12,3) = (12 × 11 × 10) ÷ (3 × 2 × 1) = 1,320 ÷ 6 = 220 possible committees.

What this means: The ordered count is exactly 6× the unordered count, because every 3-person committee can be arranged into 3! = 6 different officer lineups. Whenever two counts for the same n and r differ by a factor of r!, that factor is the number of orderings — the signature relationship nPr = nCr × r!.

Formula & methodology

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

How permutations and combinations are counted

Both count selections from n items; the difference is whether the order of the selection matters.

Permutations (order matters): nPr = n! ÷ (n − r)! — fill r slots one at a time, with n, n−1, … choices.
Combinations (order irrelevant): nCr = n! ÷ (r! × (n − r)!) — the permutation count divided by the r! ways to reorder each selection.
Cancel before multiplying: C(8,3) = (8 × 7 × 6) ÷ 6 = 56, no giant factorials needed.
Symmetry shortcut: C(n,r) = C(n,n−r) — choosing 3 from 10 equals choosing which 7 to leave out.

Example: a 6/49 lottery has C(49,6) = 13,983,816 combinations — the familiar ‘1 in 14 million’ odds.

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.

How do I know whether my problem is nPr or nCr?

Ask whether swapping two chosen items creates a different outcome. Electing a president and secretary from 10 people is P(10,2) = 90, because Alice-president/Bob-secretary differs from the reverse. Picking any 2-person committee is C(10,2) = 45, because the pair is the same regardless of order. ‘Arrangements,’ ‘rankings’ and ‘passwords’ signal permutations; ‘teams,’ ‘committees’ and ‘hands’ signal combinations.

Why does the combination formula divide by r!?

Because every unordered selection of r items can be arranged in r! different orders, and nPr counts all of those orders separately. Dividing by r! collapses each set of r! orderings back into one selection. For r = 3, each committee appears 6 times in the permutation count, so nCr = nPr ÷ 6.

What does 0! mean, and why is C(n,0) = 1?

By convention 0! = 1 — there is exactly one way to arrange nothing. Then C(n,0) = n! ÷ (0! × n!) = 1: there is exactly one way to choose an empty selection. It keeps the formulas consistent at the boundary, which matters in probability sums and the binomial theorem.

How do permutations relate to probability?

Probability is (favourable outcomes) ÷ (total outcomes), and counting gives you both numbers. The chance of being dealt a specific poker hand is 1 ÷ C(52,5) = 1/2,598,960. Use nCr when the event doesn’t care about order and nPr when it does — mixing them up is the most common counting error in probability problems.

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