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Permutation and Combination Calculator

Math

Calculate permutations (nPr) and combinations (nCr) instantly. Enter n and r to find arrangements and selections with full formula breakdown.

Reviewed by the thecalcu.com team · Last updated June 19, 2026

Total Items (n)
020
Items Chosen (r)
020

Permutations (nPr) — order matters

10P3 = 10! / (10−3)!
0

Combinations (nCr) — order doesn't matter

10C3 = 10! / (3! × (10−3)!)
0

Factorials Used

10!1
3!1
(10−3)! = 7!1

What is a nPr nCr?

The Permutation and Combination Calculator computes nPr (permutations) and nCr (combinations) for any total number of items n and selection size r, instantly returning both the number of ordered arrangements and the number of unordered selections. Permutations count arrangements where order matters, like ranking the top 3 finishers out of 10 racers, while combinations count selections where order doesn't matter, like choosing 3 people for a committee out of 10 candidates. This distinction trips up more students and professionals than almost any other concept in combinatorics, and getting it backwards silently produces the wrong answer in probability, statistics, and computer science problems. Enter n and r once and this tool gives you both figures side by side so you can see exactly how much larger nPr is than nCr for the same inputs.

Why Use a Permutation and Combination Calculator?

Manually computing factorials for anything beyond single digits gets error-prone fast, 10! alone is 3,628,800, and the arithmetic for nPr or nCr involves dividing large factorials against each other, which is easy to fumble by hand or even in a basic calculator app. This tool is built for two recurring situations: verifying homework or exam answers in a statistics or discrete math course, and quickly working out real-world counting problems like "how many ways can I arrange 5 books on a shelf" or "how many 4-person teams can I form from 12 people." Both come up often enough, in classrooms, coding interview prep, and probability puzzles, that having instant, correct answers saves real time and catches silly arithmetic slips before they compound into a wrong final answer.

How to use this nPr nCr calculator

  1. Set "Total Items (n)" to the size of your full set, for example, 10 people or 10 objects.
  2. Set "Items Chosen (r)" to how many you're selecting or arranging from that set.
  3. Read the "Permutations nPr" result if the order of selection matters (rankings, arrangements, sequences).
  4. Read the "Combinations nCr" result if only the selection matters, not the order (committees, groups, unordered picks).
  5. Adjust n or r using the sliders to explore how quickly both values grow, try increasing r toward n to see the gap between nPr and nCr shrink.

Formula & Methodology

Permutations: P(n, r) = n! / (n − r)!
Combinations: C(n, r) = n! / [r! × (n − r)!]

Where:
- n = total number of items in the set
- r = number of items being chosen or arranged
- n! = n factorial, the product of all positive integers up to n (with 0! = 1)

Worked example: For n = 10, r = 3:

P(10, 3) = 10! / 7! = 10 × 9 × 8 = 720 permutations
C(10, 3) = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120 combinations

Notice that C(10,3) = P(10,3) / 3!, confirming the general relationship nCr = nPr / r!.

Common Mistakes to Avoid

  • Applying nPr when the problem doesn't care about order. If the question asks "how many ways to pick a committee," that's nCr, using nPr will overcount by a factor of r!.
  • Forgetting that 0! = 1. This trips up manual calculations at r = 0 or r = n, where people sometimes assume the factorial term vanishes to 0 instead of 1.
  • Confusing "combination lock" language with mathematical combinations. A lock code where order matters is a permutation problem despite the name.
  • Entering r greater than n. You can't choose or arrange more items than exist in the set, the result is undefined, not a large number.
  • Mixing up which value is n and which is r. n is always the total pool size; r is always the smaller selection size, and swapping them gives a nonsensical or undefined result.

Quick Reference

n r nPr nCr
5 2 20 10
6 3 120 20
10 3 720 120
10 5 30,240 252

If you're working through a broader probability problem, follow up with the Probability Calculator or the Statistics Calculator for related distribution and summary-stat work. For number-theory adjacent problems, the GCF and LCM Calculator is a common next stop.

Frequently Asked Questions

Order matters for permutations and doesn't for combinations. Arranging 3 books on a shelf out of 5 (order matters, each arrangement is distinct) is a permutation problem, while picking 3 books to take on vacation out of 5 (the selection is what counts, not the order you pack them) is a combination problem.
By definition, there's exactly one way to arrange zero items, doing nothing, so 0! = 1 keeps the permutation and combination formulas consistent at the boundary case. It's not an arbitrary convention; it's what makes nCr = n!/(r!(n-r)!) work correctly when r = 0 or r = n.
No, you can't arrange or choose more items than you have available, so both nPr and nCr are undefined (or conventionally zero) when r > n. This calculator caps r at 20 and n at 20, and the result will show 0 if r exceeds n.
Factorials grow explosively, 20! is already over 2.4 quintillion, so values beyond that quickly exceed what's practical to display or even compute precisely with standard floating-point arithmetic. If you need larger values, most programming languages have arbitrary-precision integer libraries built for exactly this.
When r equals n, nPr simplifies to just n! since you're arranging every item with none left over. For n = 5, that's 5! = 120 total arrangements, and it's also the maximum possible value of nPr for a given n.
No, and this trips people up constantly, a combination lock cares about the order you enter digits, which makes it a permutation problem, not a combination. Mathematically it should probably be called a 'permutation lock,' but the naming stuck long before anyone worried about the distinction.
Probability problems often reduce to counting favorable outcomes divided by total possible outcomes, where both counts come from nPr or nCr depending on whether order matters in the scenario. For card and lottery problems specifically, combinations are almost always the right tool since drawing a hand doesn't care about the order the cards came out.
nPr = nCr × r!, which makes sense once you see it: for every unordered selection of r items (a combination), there are r! ways to arrange those same items in order (permutations). So combinations are always smaller than or equal to permutations for the same n and r.
No, permutations and combinations are only defined for non-negative integers because they're counting discrete arrangements or selections of physical or countable items. This calculator restricts both inputs to whole numbers between 0 and 20 for that reason.
Each entry in Pascal's Triangle is literally nCr for that row and position, which is why the triangle is such a common visual tool for teaching binomial coefficients. The symmetry you see in the triangle (nCr = nC(n-r)) reflects the fact that choosing r items to include is equivalent to choosing n-r items to leave out.
Committee selection ('choose 4 people from 10'), password counting ('how many 4-digit PINs with no repeats'), and seating arrangement problems are the classic templates, and almost every one maps directly onto either nPr or nCr depending on whether order is part of the question. Once you can correctly identify which one applies, the arithmetic itself is usually the easy part.
Also known as
nPr nCr calculatorpermutations and combinationscombinations calculatorarrangements calculatorprobability combinations