CalcaTools

Permutation and Combination Calculator

A Permutation and Combination Calculator is a free online tool that solves nPr calculator using nPr calculator, nCr calculator, how many ways to arrange. Students, teachers, and engineers use it to check homework, prep for exams, and verify hand calculations.

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

functions Shows the working, not just the answer

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.

calculate Accurate to the spec

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.

school Free for classroom use

Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.

tips_and_updates Pair with the spoke articles

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

Formula & methodology

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

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.

Frequently asked questions

What's the difference between a permutation and a combination?
Permutations care about order (ABC differs from CBA); combinations don't (ABC = CBA). Use nPr for arrangements, nCr for selections.
How do I calculate nCr?
nCr = n!/(r!(n-r)!). For 5 choose 2 that's 120/(2 x 6) = 10.
What is a factorial?
n! is the product of all integers from 1 to n (5! = 120). By definition 0! = 1.
How are lottery odds calculated?
With combinations, since the draw order doesn't matter. Picking 6 of 49 is 49C6 = about 13.98 million outcomes.
Why is nPr larger than nCr?
Each unordered combination can be arranged in r! ordered ways, so nPr = nCr x r!.
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.