Compute nPr, nCr, and factorials exactly — with a clear breakdown of order-matters vs order-doesn't.
A permutation counts arrangements where order matters — a race podium (gold, silver, bronze) is different from (silver, gold, bronze). A combination counts selections where order does not matter — a committee of {Alice, Bob, Carol} is the same group no matter how you list them.
Because every combination of r items can be arranged in r! different orders, nPr is always r! times larger than nCr.
Permutations: nPr = n! / (n − r)!
Combinations: nCr = n! / (r! · (n − r)!)
Here n! ("n factorial") is the product of all whole numbers from 1 to n — for example 5! = 5·4·3·2·1 = 120. And 0! is defined as 1.
nCr is the binomial coefficient. It counts the number of ways to get k successes in n trials, and it's exactly the C(n,k) term in the binomial probability formula P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ.
The chart above plots all C(n,k) for k = 0…n — one full row of Pascal's triangle. If you're working with success/failure trials, head to the binomial distribution calculator.
By convention 0! = 1. There is exactly one way to arrange or choose nothing — the empty selection — so nC0 = 1 and nCn = 1. Defining 0! = 1 keeps every formula consistent.
This calculator computes the standard without-repetition nPr and nCr. If repetition is allowed — like a 4-digit PIN where digits can repeat — the count is simply nʳ (here 10⁴ = 10,000). Combinations with repetition use the formula C(n+r−1, r).