Permutation & Combination Calculator

Permutation & Combination Calculator (nPr / nCr)

This free permutation and combination calculator solves the two most fundamental counting problems in probability and combinatorics: how many ways can you arrange r items out of a total of n (permutation, nPr), and how many ways can you choose r items out of a total of n (combination, nCr). Both are built directly from factorials, and both show up constantly in probability theory, statistics, computer science, and standardized-test math — anywhere a problem asks 'in how many ways can this happen?'

The single most important distinction between the two is whether order matters. A permutation counts arrangements — assigning 1st, 2nd, and 3rd place in a race, forming a password from a set of letters, or seating people in specific chairs are all permutation problems, because swapping two items creates a genuinely different outcome. A combination counts selections — picking a 5-card poker hand, choosing a committee of 3 people from a group of 10, or selecting lottery numbers are all combination problems, because the order in which you pick the items doesn't create a different result.

Permutation Formula (nPr)

The number of ways to arrange r items, chosen from a total of n distinct items, where order matters, is:

nPr = n! / (n − r)!

Here, n! (n factorial) is the product of all positive integers up to n (n! = n × (n − 1) × (n − 2) × ... × 1), and (n − r)! is the factorial of the remaining, unchosen items. Dividing n! by (n − r)! effectively multiplies the first r terms of n! together — n × (n − 1) × ... × (n − r + 1) — which counts every distinct ordered arrangement of r items out of n.

Combination Formula (nCr)

The number of ways to choose r items out of n total items, where order does not matter, is:

nCr = n! / (r! × (n − r)!)

This is the same numerator as the permutation formula, but divided by an extra r! in the denominator — because every group of r items chosen can be arranged in r! different orders, and a combination treats all of those orders as the same outcome. In other words, nCr = nPr / r!. This value is also known as the binomial coefficient, written C(n, r) or 'n choose r', and it's the same quantity used in the binomial probability formula and in Pascal's Triangle.

How to Use This Calculator

Enter the total number of items (n) and the number of items being arranged or chosen (r), then pick whether you want the permutation, the combination, or both. The calculator instantly computes nPr and nCr using exact integer arithmetic (not floating-point approximations), so results stay precise even for fairly large n and r — along with the individual factorials n!, r!, and (n − r)! that make up the formula, and a chart showing every possible nCr value across the full range of r for the given n.

• Use nPr when the arrangement or sequence of the chosen items matters — rankings, passwords, seating charts, race finishes.
• Use nCr when only the selection matters, not the order — teams, committees, card hands, lottery number sets.
• r must always be less than or equal to n, since you can't choose or arrange more items than exist in the set.
• For very large n (factorials with dozens of digits), the calculator automatically switches to a compact scientific-notation display so the result stays readable.

Worked Examples

Permutation example: In how many ways can a 1st, 2nd, and 3rd place be awarded among 8 runners? Here n = 8 and r = 3, since order (which runner finishes 1st vs. 2nd vs. 3rd) matters. nPr = 8! / (8 − 3)! = 8! / 5! = 8 × 7 × 6 = 336 possible ways to award the three placements.

Combination example: In how many ways can a committee of 3 people be chosen from a group of 8, where the order of selection doesn't matter? Here n = 8 and r = 3 again, but now nCr = 8! / (3! × 5!) = (8 × 7 × 6) / (3 × 2 × 1) = 336 / 6 = 56 possible committees — far fewer than the 336 permutations, because each group of 3 people can be arranged in 3! = 6 different orders that all count as the same committee.

Permutation & Combination in Probability

Beyond pure counting, nPr and nCr are the building blocks of classical probability calculations. Whenever an event's probability depends on 'how many favorable outcomes out of how many total possible outcomes', those two counts are usually computed with permutations or combinations. The binomial coefficient nCr, in particular, appears directly inside the binomial probability formula, where C(n, x) counts the number of distinct ways x successes can occur among n trials — which is exactly why this calculator lives alongside the binomial, Poisson, and normal distribution calculators rather than the general statistics tools.