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Mathematics — Combinatorics

Permutation and Combination Calculator (nPr & nCr)

Free online calculator for permutations and combinations. Calculate nCr, nPr, and n choose r with step-by-step formulas — solve 5C3, 6C2, 10C3, 5 choose 3, and hundreds more notation formats instantly.

A full-featured combinatorial calculator and permutation solver for homework, probability, and statistics. Supports n choose k, x choose y, circular permutation, and ncr/npr notation with exact results up to n = 200.

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Exact n=200 Detailed Steps Advanced Modes Mode: Standard
Formula reference
Permutation
P(n, r) = n! (n - r)!
Combination
C(n, r) = n! r! x (n - r)!
Example: 10P3 = 720, 10C3 = 120

Popular nCr and nPr homework examples

Click any example to auto-fill the calculator with combinations and permutations students search for most often.

Step-by-step

Select Calculation Type

Pick how you want to count arrangements.

Enter Values

Use direct inputs or quick notation.

Supports nCr, nPr, 5 choose 3, 5 choose 4, 5c4, 6C2, 25P3 permutation, 2P4 permutation, and P(10,4).

Total items available

Number of items to select

Results

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Permutation
Ready when you are

Enter values for n and r to see formulas, steps, and the final answer.

Example: 8C3 = 56 Example: 7P2 = 42

Advanced counting modes

Use these when the problem includes repeated choices, circular seating, or repeated item groups.

Optional

nCr and nPr example answers — quick reference

Use these worked examples to check homework, probability, and statistics problems. Covers 5C3, 5C2, 5C4, 6C2, 6C3, 10C3, 9 nCr 3, 3P2, 25P3, and more.

5C3 combination value

5C3 (5 choose 3, 5choose3) = 5!/(3!·2!) = 10. The 5c3 formula counts ways to pick 3 from 5 without order.

5C2 & 5C4 combination

5C2 = 10 and 5C4 = 5. Notice C(5,2) = C(5,3) because nCr is symmetric: n!/(r!·(n−r)!).

6C2 & 6C3 calculator

6C2 (6choose2, 6nCr2) = 15. 6C3 (6 choose 3, 6choose3) = 20. Formula: 6!/(r!·(6−r)!).

10C3 combination & 10P3

10C3 = 120 (unordered). 10P3 = 720 (ordered). Ratio: nPr/nCr = r! = 6.

9 nCr 3 & 3 nCr 2

9 nCr 3 = 84. 3 nCr 2 (3 choose 2) = 3. Use the n c r formula n!/(r!·(n−r)!).

25P3 & 25C3 permutation

25P3 = 25×24×23 = 13,800. 25C3 combination = 2,300. Order matters for nPr.

5 permutation 3 & 5 permutation 5

5P3 = 60. 5P5 = 120 = 5!. When r = n, nPr equals n! because (n−r)! = 0! = 1.

4 permute 3, 4 choose 4, 10P10

4P3 = 24. 4 choose 4 = 1 (only one way). 10P10 = 3,628,800 = 10!.

nPr vs nCr — when to use which

The difference between nPr and nCr: use nPr when order matters, nCr when it does not. nPr = nCr × r!.

Permutation and combination calculator guide

How to calculate permutations and combinations with nPr and nCr

Permutation and combination problems are about counting possibilities without listing every outcome by hand. A permutation (nPr) counts arrangements where order matters — ranking winners, creating passcodes, or seating people in a sequence. A combination (nCr) counts selections where order does not matter — choosing committee members, drawing lottery numbers, or selecting ingredients. This calculator with combinations and permutations supports nCr, nPr, n choose r, n choose k, and x choose y notation with advanced modes so you can move from homework examples to real-world counting problems.

The key decision is whether changing the order creates a new outcome. If ABC and ACB are different, use a permutation solver. If they represent the same group, use a combination formula calculator. The formula for nPr is n!/(n−r)!, and the formula for nCr is n!/(r!·(n−r)!). Factorials are the foundation: n! multiplies every positive integer from n down to 1, and 0! equals 1. For larger values — like calculating 60 choose 4 or evaluating expressions such as (6C2−2)÷(2C2) — exact integer handling and simplified multiplication are critical because factorials grow extremely quickly. You can also calculate number of combinations using the identity C(n,r) = C(n, n−r).

Core features

  • Standard nPr and nCr calculations with the full npr formula and ncr formula shown step by step.
  • Quick notation input: type 5C3, 6c2, 8nCr3, 5ncr3, 5 choose 4, 5choose2, 6choose3, 2p4, or P(10,4) and the calculator auto-fills.
  • Advanced repetition mode for permutations and combinations with repetition — ideal for PIN codes and repeated selections.
  • Circular permutation calculator mode for round-table and loop arrangements using (n−1)! formula.
  • Multiset mode for repeated items such as letters in MISSISSIPPI.
  • Copyable results, formulas, and shareable calculation links. Works as a complete npr ncr calculator and pnc calculator.

How to use the nPr and nCr calculator

Step 1: Decide whether order matters. Choose Permutation (nPr) when order matters and Combination (nCr) when it does not. This is the core difference between nPr and nCr.

Step 2: Enter n (total items) and r (items to select), or type quick notation like 5C3, 10C3, 5 choose 2, n choose r, or 25P3 directly.

Step 3: For special problems, select an advanced mode: Repetition, Circular permutation, or Multiset.

Step 4: Review the formula, step-by-step calculation, and final answer. Copy the result or share the link. Evaluate nPr with examples like 14P2.

Professional use cases

Probability and statistics

Use nCr probability formula and nPr formula to count outcomes before calculating event probability in cards, dice, lotteries, sampling, or random draws.

Exam and homework checking

Verify answers for problems like 5c3 calculator, 6c3 combination, 3p2 combination, 5 pick 2, and 4c3 combination with step-by-step explanations.

Scheduling and ranking

Use the permutation formula calculator to count possible orders for winners, tasks, presentations, playlist sequences, and seating arrangements.

Combinatorial analysis

Calculate number combinations for large sets — 24C4 combination, 25C3, 60 choose 4 — or evaluate compound expressions like 3C3−6C2. The combinatorics calculator handles exact integer results up to n=200.

Tips for better results

  • Ask whether swapping two selected items changes the result; if yes, use the npr permutation formula.
  • For combinations, C(n, r) = C(n, n−r). So 5 choose 2 = 5 choose 3 = 10.
  • Use repetition mode when the same item can be selected more than once — the formula becomes n^r for permutations.
  • Use circular mode when rotations of the same arrangement should not be counted again — the circular permutation calculator Pn uses (n−1)!.
  • Very large factorial results grow quickly; copy the exact result when precision matters. The calculator handles (n−r)! efficiently.

Permutation and Combination — detailed FAQs

What is the difference between permutation and combination?

A permutation counts ordered arrangements (nPr), while a combination counts unordered selections (nCr). If changing the order changes the outcome, use the permutation solver. Our calculator for permutations and combinations labels each formula clearly.

What does nCr mean and what is the ncr formula?

nCr means the number of ways to choose r items from n when order does not matter. The ncr formula is n!/(r!·(n−r)!). Also written as 'n choose r', 'combinations nCr', or 'ncr combination'. Use our ncr calculator to calculate nCr for any values.

What does nPr mean and what is the npr formula?

nPr means the number of ordered arrangements of r items from n. The npr formula is n!/(n−r)!. Also called 'permutations nPr' or 'npr permutation'. The npr equation differs from nCr by the r! factor. Use our n p r calculator for instant results.

How do I use nCr on a calculator?

Enter n and r, then select Combination (nCr). Or type notation like 5C3, 8nCr3, or 5 choose 3 in the quick-notation field. The ncr function calculator auto-fills and shows the step-by-step ncr calculation.

What is 5C3 value and how do I calculate 5C3?

5C3 value is 10. The 5c3 formula is 5!/(3!·2!) = 120/(6·2) = 10. You can also write it as 5 choose 3, 5choose3, or 5choose 3. Use our 5c3 calculator to verify.

When to use nPr and when to use nCr?

Use nPr when the sequence of selected items matters — seating, rankings, passcodes. Use nCr when only the group matters — committees, lottery draws, card hands. This is the core question of 'npr or ncr' and 'ncr v npr' comparisons.

Why is 0 factorial equal to 1?

0! is defined as 1 because it represents the number of ways to arrange nothing: one empty arrangement. This keeps the factorial formula n!/(r!·(n−r)!) consistent when r = 0 or r = n.

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