What is the difference between nCr and nPr?
nCr counts unordered selections (committees, hands). nPr counts ordered arrangements (passwords, podiums). For the same n and r, nPr is always ≥ nCr.
10C3 = 120
Count unordered ways to choose r items from n using nCr = n!/(r!(n−r)!)—calculated locally in your browser. Compare with permutations when order matters.
10C3 = 120
A combination calculator counts unordered selections: nCr = n! / (r!(n−r)!)—how many ways to choose r items from n when order does not matter.
Combinations answer counting questions where order is irrelevant: lottery lines, committees, card hands, and sample subsets. The binomial coefficient nCr (also written C(n,r) or "n choose r") equals n factorial divided by r factorial times (n−r) factorial.
Contrast with permutations nPr = n!/(n−r)!, which count ordered arrangements (PIN codes, race podiums). If choosing {A,B} is the same as {B,A}, use nCr; if ABC differs from CBA, use nPr.
EverydayTools computes nCr with arbitrary-precision integers in your browser for n up to 500—inputs stay on your device.
Use nCr when order does not matter; use nPr when order matters.
Concise answers for common searches — definitions, steps, and comparisons.
nCr counts unordered selections (committees, hands). nPr counts ordered arrangements (passwords, podiums). For the same n and r, nPr is always ≥ nCr.
You cannot choose more distinct items than exist in the pool; nCr is defined for 0 ≤ r ≤ n.
No. nCr is calculated locally in your browser with JavaScript big integers.
n is the total number of distinct items or trials in the pool (non-negative integer).
r is how many items you choose at once; require 0 ≤ r ≤ n.
The tool displays nCr = n!/(r!(n−r)!) using big integers—for example 10C3 = 120.
If order matters (rankings, passwords), use the permutation calculator for nPr instead.
Input
n = 10 people, r = 3 seatsOutput
10C3 = 120 committeesOrder of the three members does not matter—only who is on the committee.
Input
n = 52 cards, r = 5Output
52C5 = 2,598,960 handsStandard count of unordered 5-card deals from a full deck.
Common real-world scenarios where this tool saves time.
Verify nCr by hand or calculator before submitting counting exercises.
Count possible unordered ticket combinations or loot pools.
Determine how many subsets of size r exist from a population of n.
| Tool | Formula | Order matters? | Example |
|---|---|---|---|
| Combination (nCr) | n!/(r!(n−r)!) | No | 10C3 = 120 committees |
| Permutation (nPr) | n!/(n−r)! | Yes | 10P3 = 720 lineups |
| Probability calculator | P(A), P(A∩B) | N/A | Chance of independent events |
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nCr (n choose r) counts unordered selections: nCr = n! / (r!(n−r)!). Example: 5C2 = 10 ways to pick 2 items from 5.
Use nCr when order does not matter (choosing a committee). Use nPr when order matters (podium places, PIN digits).
By convention, 0C0 = 1 (one way to choose nothing from nothing). The tool follows standard combinatorial definitions for edge cases.
Factorials grow extremely fast; n > 500 can exceed practical browser memory for exact integer arithmetic. Use smaller n or specialized software for huge values.
This tool counts arrangements (how many ways). The probability calculator computes event likelihoods between 0 and 1 for independent events.
No. n and r are processed locally in your browser—they are not sent to EverydayTools servers.
Values n and r are processed in your browser—they are not uploaded to EverydayTools servers.
Uses exact integer arithmetic for nCr when n ≤ 500 and 0 ≤ r ≤ n.
Educational combinatorics aid—not a substitute for formal statistical consulting.
More free tools for the same workflow.
Calculate factorial n! for 0 ≤ n ≤ 500 with exact big-integer results. No upload—runs locally in your browser. Free, instant n! for combinatorics.
Calculate permutations nPr = n!/(n−r)! when order matters—5P2, 10P3, 26P3 & more. Runs locally in your browser, no upload. Exact results up to n=500.
Calculate P(A), complements, P(A and B), and P(A or B) for independent events. No upload—runs locally in your browser. Free, instant.
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Reviewed by EverydayTools Editorial Team on 2026-05-20.