Permutation and Combination Calculator

How many ways can you arrange or choose items? Permutations count ordered arrangements; combinations count unordered selections. This calculator handles both — nPr and nCr — instantly.

How to Use This Calculator

1. Enter n (total number of items).
2. Enter r (number of items to choose or arrange).
3. Select Permutation (nPr) or Combination (nCr).
4. Click Calculate to get the result with step-by-step factorials shown.

Formulas

Permutation (order matters): P(n, r) = n! / (n−r)!

Combination (order doesn't matter): C(n, r) = n! / [r! × (n−r)!]

The key difference: permutations count ABC and BAC as different arrangements. Combinations count them as the same selection.

Example Calculations

Permutation: How many ways can 3 people finish 1st, 2nd, 3rd out of 8 runners?
P(8, 3) = 8! / 5! = 8 × 7 × 6 = 336

Combination: How many ways can you choose 3 committee members from 8 candidates?
C(8, 3) = 8! / (3! × 5!) = 56 ÷ 6 = 56

The committee has 56 possible compositions; the race finish has 336 possible orderings — same numbers, very different counts.

When to Use Permutations vs. Combinations

• Permutations: Passwords, license plates, race finishing orders, arranging books on a shelf, picking officers (President, VP, Secretary from a club).
• Combinations: Lottery tickets, committee selection, card hands, choosing pizza toppings, survey response subsets.

Common Mistakes to Avoid

• Using permutation when order doesn't matter — A lottery picks 6 numbers from 1–49. 1,2,3,4,5,6 and 6,5,4,3,2,1 are the same ticket → combination, not permutation.
• Forgetting r cannot exceed n — You can't choose more items than exist. P(5,6) and C(5,6) are both undefined/zero.
• Computing huge factorials manually — Use this calculator. 20! = 2,432,902,008,176,640,000 — impractical to compute by hand.

Frequently Asked Questions

What is the difference between permutation and combination?

Permutations: order matters (ABC ≠ BAC). Combinations: order does not matter (ABC = BAC = CAB). Every combination corresponds to r! permutations, so nPr = r! × nCr.

What is a factorial?

n! = n × (n−1) × (n−2) × ... × 2 × 1. By convention, 0! = 1. Factorials grow explosively: 10! = 3,628,800 and 20! ≈ 2.4 × 10¹⁸.

How is nCr used in the binomial theorem?

The coefficient of xʳ in the expansion of (x+y)ⁿ is C(n,r). Pascal's triangle is just a table of combinations — C(n,r) = C(n−1,r−1) + C(n−1,r).

Conclusion

Counting arrangements and selections is fundamental to probability, statistics, and discrete mathematics. Use this calculator for any combinatorics problem — and always ask yourself first: does the order matter?

Related: Probability Calculator | Statistics Calculator | Random Number Generator