Permutation and Combination Calculator

Permutations and combinations are counting techniques in combinatorics. A permutation P(n, r) counts the number of ways to arrange r items chosen from n distinct items where order matters. A combination C(n, r) counts the number of ways to select r items from n distinct items where order does not matter. The key difference: choosing {A, B, C}and {C, B, A} is the same combination but two different permutations. Our permutation combination calculator computes both simultaneously using exact BigInt arithmetic for n up to 1000.

1. 1. Enter n and r: n is the total number of items; r is the number of items being chosen or arranged. Both must be non-negative integers with r ≤ n and n ≤ 1000.
2. 2. Instant calculation: Click Calculate to see P(n,r) and C(n,r) side by side, along with the relationship P(n,r) = C(n,r) × r! — all processed locally in your browser.
3. 3. View steps and Pascal's Triangle: Click "Show Steps" for the full formula derivation. For n ≤ 20, a Pascal's Triangle row shows all C(n, k) values with your selected r highlighted.

• Permutation P(n, r): P(n, r) = n! / (n − r)! The number of ordered arrangements of r items from n distinct items. Example: P(5, 2) = 5! / 3! = 5 × 4 = 20. There are 20 ways to arrange 2 items chosen from 5 (e.g. first and second place in a race of 5 runners).
• Combination C(n, r): C(n, r) = n! / (r! × (n − r)!) The number of unordered selections of r items from n distinct items. Also written as ⁿCᵣ or "n choose r". Example: C(5, 2) = 5! / (2! × 3!) = 10. There are 10 ways to choose 2 items from 5 (e.g. selecting 2 players from a team of 5).
• Relationship Between P and C: P(n, r) = C(n, r) × r! Every combination can be arranged in r! ways to produce a permutation. So P(n,r) is always r! times larger than C(n,r). For r = 0 or r = 1, P = C. For r = n, P(n,n) = n! and C(n,n) = 1.

• Symmetry: C(n, r) = C(n, n−r). Choosing r items is equivalent to choosing which n−r items to leave out.
• Edge cases: C(n, 0) = C(n, n) = 1. P(n, 0) = 1. P(n, n) = n!
• Pascal's identity:C(n, r) = C(n−1, r−1) + C(n−1, r). This is the rule that generates Pascal's Triangle.
• Sum of a row: The sum of all C(n, r) for r = 0 to n equals 2ⁿ — the total number of subsets of n items.