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Combinations & Permutations Calculator

Compute nCr and nPr — with or without replacement. Choose order-matters or order-doesn't-matter.

When to use which?

  • Combinations — order doesn't matter. Example: picking 3 toppings from 8 (a "sandwich combinations" problem).
  • Permutations — order matters. Example: arranging 3 books from 10 on a shelf.
  • With replacement — items can be reused. Example: rolling a 6-sided die 4 times → 6⁴ outcomes.

Combinations & permutations formulas

Combinations (order doesn’t matter)C(n, r) = n!r! (n − r)!
Permutations (order matters)P(n, r) = n!(n − r)!
Permutations with repetitionnr
Combinations with repetitionC(n + r − 1, r) = (n + r − 1)!r! (n − 1)!
Link between themP(n, r) = C(n, r) × r!

Where n = items to choose from, r = items chosen

Worked examples

  1. Choose a committee of 3 from 10 people.

    1. C(10, 3) = 10 × 9 × 83 × 2 × 1 = 7206

    Answer: 120 committees

  2. Choose a president, VP and secretary from 10 people.

    1. Order matters: P(10, 3) = 10 × 9 × 8

    Answer: 720 ways

  3. Pick 6 numbers from 49 in a lottery.

    1. C(49, 6) = 49! ÷ (6! × 43!)

    Answer: 13,983,816 — 1 in ~14 million

  4. How many 4-digit PINs are there?

    1. Repetition allowed, order matters: 104

    Answer: 10,000

  5. 3 scoops of ice cream from 5 flavors (repeats allowed, order doesn’t matter).

    1. C(5 + 3 − 1, 3) = C(7, 3)

    Answer: 35

Tips & common mistakes

  • Ask "does order matter?" — if swapping two picks gives a new outcome, use permutations.
  • Words with repeated letters: divide by the factorial of each repeat — MISSISSIPPI = 11! ÷ (4! 4! 2!) = 34,650.