What is the difference between permutation and combination?

Permutations count ordered arrangements: P(n,r) = n!/(n−r)!. Combinations count unordered selections: C(n,r) = n!/(r!(n−r)!). Order matters for permutations (AB≠BA), not for combinations (AB=BA).

When should I use nCr vs nPr?

Use nPr when order matters (e.g. first/second/third place in a race). Use nCr when order doesn't matter (e.g. selecting a team of 5 from 20 players, choosing lottery numbers).

Back to home

Permutation & Combination Calculator

Calculate P(n,r) and C(n,r) with factorial working and Pascal's triangle row.

Inputs

P(n,r) = n!/(n−r)! | C(n,r) = n!/(r!(n−r)!)

n (total items)

r (chosen items)

Permutations vs. Combinations

A permutation is an arrangement where order matters. P(n,r) counts the number of ways to arrange r items from n distinct items in a specific order. A combination is a selection where order doesn't matter — C(n,r) counts the number of ways to choose r items from n without regard to order.

Real-World Examples

Permutations: arranging 3 books from a shelf of 10, assigning gold/silver/bronze medals. Combinations: choosing a team of 3 from 10 players, picking lottery numbers. Remember: P(n,r) = C(n,r) × r! because every combination can be arranged in r! ways.

Pascal's Triangle

Pascal's triangle displays all combinations for a given n in a single row. Row n contains C(n,0), C(n,1), …, C(n,n). The values in any row sum to 2ⁿ and are the binomial coefficients used in the binomial theorem.

Related Calculators

Explore more free online calculators to help with your calculations

Probability Calculator

Calculate event probabilities

Sample Size Calculator

Required sample sizes

Statistics Calculator

Full descriptive statistics

Take this calculation further

Browse the math calculators collection for more tools in this subject.

For a related calculation, use Z-Score Calculator to convert values to z-scores and percentiles, or find the percentile for any z-score using the standard normal distribution. Alternatively, use Probability Calculator to calculate probabilities for single events, two independent events, and conditional probability.

For the underlying method, read Understanding Standard Deviation and Why It Matters or A Practical Guide to Statistical Sample Size: How Many People Do You Actually Need to Survey?.