Combinatorics Calculator – nCr, nPr & More

Counting problems differ on two axes — does order matter, and can items repeat — producing four standard formulas the calculator computes side by side. Choosing 2 from 5: unordered without repetition gives C(5,2) = 10; ordered without repetition gives P(5,2) = 5×4 = 20; ordered with repetition gives 5² = 25; unordered with repetition gives C(6,2) = 15. The tool prints all four together with the formula substitution so the source of each number is visible, then helps you classify your problem with two questions (order? repeats?). Factorial expansions are shown canceled, e.g. C(10,3) = (10×9×8)/(3×2×1) = 120, avoiding huge intermediate factorials. Sum rules appear for completeness: ΣC(n,k) = 2ⁿ counts all subsets, so a 5-element set has 32 subsets — the answer to many "how many groups" questions.

1. 1

   Enter n and r

   The pool size and selection size

2. 2

   Answer two questions

   Does order matter? Are repeats allowed?

3. 3

   Read the matching formula

   All four counts are shown with substitutions

4. 4

   Sanity-check the spread

   Ordered ≥ unordered, repetition ≥ no-repetition

• 1

  Order matters → permutations; otherwise combinations

• 2

  Repetition allowed → nʳ (ordered) or C(n+r−1, r) (unordered)

• 3

  All subsets of an n-set total 2ⁿ

• Using nPr when the problem says "group" or "committee" — order-free

• Forgetting repetition when items are returned or unlimited

• Computing C(5,2) as 20 by skipping the r! divisor

• Confusing 2ⁿ subsets with n! arrangements