Combinations & Permutations Calculator
Calculate combinations and permutations.
Input
Result
What is a Combinations & Permutations Calculator?
A Combinations & Permutations Calculator computes nCr (combinations) and nPr (permutations) for any set size n and selection size r. It is essential for probability, statistics, counting problems, and combinatorial math where you need to know how many ways you can select or arrange items.
How to calculate combinations and permutations manually (the formula)
Combinations (order does not matter): nCr = n! / (r! × (n − r)!). Permutations (order matters): nPr = n! / (n − r)!. The exclamation mark denotes the factorial function — multiply all whole numbers from n down to 1.
Example calculation
How many ways can you choose 2 toppings from a menu of 5? This is a combination: 5C2 = 5! / (2! × 3!) = 120 / (2 × 6) = 10. How many ways can 3 runners place 1st, 2nd, 3rd out of 8? This is a permutation: 8P3 = 8! / 5! = 40320 / 120 = 336.
Common mistakes
- Using the wrong formula — Use permutations when order matters; combinations when it does not. Ask: does swapping the items give a different result?
- Forgetting that 0! = 1 — The factorial of zero is 1, not 0. This matters when r = n or r = 0.
- Confusing n and r — n is the pool you select from; r is how many you pick. n is always greater than or equal to r.
- Miscounting with large numbers — Factorials grow extremely fast. Use the calculator to avoid arithmetic errors.
Frequently asked questions
What is the difference between combinations and permutations?
Combinations (nCr) count selections where order does not matter. Permutations (nPr) count arrangements where order matters. For example, choosing 3 letters from ABC gives 1 combination (ABC) but 6 permutations (ABC, ACB, BAC, BCA, CAB, CBA).
How do I calculate combinations?
The formula for combinations is nCr = n! / (r! × (n − r)!). Enter your n and r values and the calculator applies this formula instantly.
How do I calculate permutations?
The formula for permutations is nPr = n! / (n − r)!. Enter your n and r values and the calculator computes nPr for you.
What does n and r mean in nCr and nPr?
n is the total number of items to choose from. r is the number of items you are selecting. For example, with n=5 and r=3 you are choosing 3 items from a set of 5.
Can I use this calculator for probability problems?
Yes. Many probability calculations rely on combinations and permutations. The nCr result tells you the total number of equally likely outcomes, which you can use to compute probabilities.
What is a practical example of combinations vs permutations?
A lock combination is actually a permutation because order matters (1-2-3 is different from 3-2-1). Choosing a committee from a group is a combination because the team is the same regardless of selection order.