Free nPr and nCr Calculator
Enter values for n (total items) and r (items selected). Choose whether order matters and whether repetition is allowed.
What This Calculator Does
This permutations and combinations calculator quickly computes nPr and nCr values for counting problems in math, probability, statistics, and real-life decision making. You can calculate results with or without repetition and switch between permutation mode (order matters) and combination mode (order does not matter).
Permutation vs. Combination
Permutation (nPr): order matters
Use permutations when different arrangements count as different outcomes. For example, assigning gold, silver, and bronze medals to 3 people out of 10 is a permutation because positions are different.
- Without repetition: nPr = n! / (n - r)!
- With repetition: nr
Combination (nCr): order does not matter
Use combinations when you only care about which items are selected, not their order. Choosing 3 committee members out of 10 is a combination.
- Without repetition: nCr = n! / (r!(n - r)!)
- With repetition: (n + r - 1)! / (r!(n - 1)!)
Quick Examples
Example 1: Password slots
If you pick 4 characters from 10 choices and character order matters, use permutation. If repetition is allowed, the count is 104 = 10,000.
Example 2: Team selection
If you choose 5 players from 12 and order does not matter, use combination: nCr = 12C5 = 792.
When to Use Which Formula
- Use nPr when positions, ranks, or sequence are important.
- Use nCr when selecting a group regardless of arrangement.
- Use repetition only when the same item can appear more than once.
Common Mistakes to Avoid
- Mixing up order matters vs. order does not matter.
- Using "without repetition" when r is greater than n (this is not valid).
- Forgetting that combinations with repetition use a different formula.
Applications in Real Life
Permutation and combination calculations are useful in:
- Probability and statistics problems
- Lottery and game odds
- Scheduling and seat arrangements
- Product bundle planning and offer design
- Cryptography and code-space estimation
FAQ
Can n and r be zero?
Yes. In many counting contexts, selecting zero items has exactly one outcome (the empty selection), so several formulas return 1 in that case.
Why are my results so large?
Factorials and exponential growth become huge very quickly. This calculator uses big-integer arithmetic so it can display exact large values.
Do I need to simplify manually?
No. The calculator computes exact integer outputs automatically.