Permutation & Combination Calculator nPr & nCr Calc

What Is the Permutation and Combination Calculator?

The permutation and combination calculator is a free online tool that computes the number of possible arrangements (nPr) and selections (nCr) from a set of items. In mathematics, permutations count ordered arrangements — where the sequence matters — while combinations count unordered selections — where only the group membership matters. Understanding the difference between permutation and combination is essential in probability, statistics, competitive exams, and everyday decision-making.

Simply enter the total number of items (n) and the number you want to choose (r), and the calculator returns nPr, nCr, all intermediate factorials, and a step-by-step breakdown of each calculation.

Key Features

• Instant nPr & nCr: Both permutation and combination values are calculated simultaneously.
• Full factorial display: See n!, r!, and (n − r)! alongside the main results.
• Step-by-step workings: A detailed breakdown shows exactly how the answer is derived.
• Large-number support: Handles values of n up to 170 (the limit of standard floating-point precision).
• Input validation: Automatically ensures r ≤ n and both values are non-negative integers.
• No sign-up required: 100% free with no account or download needed.

Permutation & Combination Formulas — How It’s Calculated

How to Use — Step-by-Step

1. Enter n (total items): Type the total number of items in the set. For example, enter 10 if you have 10 players.
2. Enter r (items chosen): Type how many items you are selecting or arranging. For example, enter 3 to choose 3 players.
3. Click “Calculate”: The calculator instantly displays nPr, nCr, all factorials, and the step-by-step workings.
4. Review the step-by-step section: Scroll down to see the full factorial expansion and cancellation used to arrive at the answer.

Practical Examples with Real Numbers

• Example 1 — Race podium: 8 runners compete for gold, silver, and bronze. How many possible podium arrangements? nPr = 8! ÷ (8 − 3)! = 8! ÷ 5! = 8 × 7 × 6 = 336 arrangements.
• Example 2 — Lottery ticket: Pick 6 numbers from 49 (order does not matter). nCr = 49! ÷ (6! × 43!) = 13,983,816 possible combinations.
• Example 3 — Committee selection: Choose a 4-person committee from 12 candidates. nCr = 12! ÷ (4! × 8!) = 495 ways.

Real-World Use Cases

• Exam & interview scheduling: Determine the number of ways to assign time slots to candidates (permutation).
• Lottery & games of chance: Calculate the odds of winning by finding total combinations.
• Password & PIN generation: Count possible passwords of a given length from a character set.
• Sports tournaments: Figure out how many unique match-ups or team rosters are possible.
• Project management: Determine the number of ways to assign tasks to team members.
• Genetics & biology: Calculate gene or allele combinations in inheritance studies.

Understanding Your Results

After clicking “Calculate”, the results card shows:

• nPr (Permutation): The number of ordered arrangements. This value is always ≥ nCr.
• nCr (Combination): The number of unordered selections. This is nPr divided by r!.
• n!, r!, (n − r)!: The intermediate factorial values used in both formulas.
• Step-by-step: A detailed breakdown showing the factorial expansion, cancellation, and final multiplication.

Tips & Best Practices

• Ask “Does order matter?”: If rearranging the same items gives a different outcome, use permutation. If not, use combination.
• Start with combination: When in doubt, calculate nCr first. If order matters, simply multiply by r! to get nPr.
• Check your constraint: r must be ≤ n. If r > n, there are zero ways to choose or arrange.
• Use symmetry for nCr: nCr = nC(n − r). Choosing 2 from 10 is the same count as choosing 8 from 10.
• Watch for repetition: nPr and nCr assume items are distinct and chosen without replacement. If repetition is allowed, use n^r instead.

Common Mistakes to Avoid

• Using permutation when combination is needed (or vice versa): Choosing a 3-person team is a combination; assigning captain, vice-captain, and secretary is a permutation.
• Forgetting that 0! = 1: Many students mistakenly use 0! = 0, which leads to division-by-zero errors.
• Confusing “with repetition” and “without repetition”: Standard nPr and nCr assume no repetition. Use different formulas when items can repeat.
• Overflowing large factorials: Factorials grow extremely fast. This calculator supports up to n = 170; beyond that, results exceed standard number limits.

Frequently Asked Questions

• What is the difference between permutation and combination?

  Permutation (nPr) counts arrangements where order matters. Combination (nCr) counts selections where order does not matter. For the same n and r, nPr is always ≥ nCr because nPr = nCr × r!.

• What is a factorial?

  A factorial (n!) is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By mathematical convention, 0! = 1.

• Can r be greater than n?

  No. You cannot choose or arrange more items than are available. If r > n, both nPr and nCr equal 0.

• How is this calculator useful for probability problems?

  Many probability questions require counting favorable and total outcomes. Permutations and combinations provide those counts, which you then plug into the probability formula P = favorable ÷ total.