This combination calculator computes combinations (nCr — order does not matter) and permutations (nPr — order matters) for any values of n and r, showing the factorial steps so you can check homework or settle “how many ways” questions instantly.

Combinations C(n, r)–
Permutations P(n, r)–
n! (factorial)–

Combinations vs. Permutations

Both count ways to choose r items from n — the difference is whether order matters:

  • Combination (nCr): order does NOT matter. A committee of 3 from 10 people is the same committee however you list it: C(10, 3) = 120 (verified).
  • Permutation (nPr): order DOES matter. Gold-silver-bronze from 10 athletes has P(10, 3) = 720 outcomes (verified) — 6 orderings (3!) of each podium set.

Ask one question to choose correctly: “If I swap two chosen items, is it a different result?” A swapped lock code is different (permutation); a swapped pizza topping is not (combination). This calculator belongs to our Statistics Calculators collection.

The Formulas

  • C(n, r) = n! ÷ (r! × (n − r)!)
  • P(n, r) = n! ÷ (n − r)!
  • n! (“n factorial”) = n × (n−1) × … × 1 — e.g., 10! = 3,628,800 (verified)

Factorials explode quickly, which is why this calculator uses exact big-number arithmetic — results stay precise far beyond ordinary calculator limits (up to n = 500).

Famous Combination Numbers

  • Lottery 6/49: C(49, 6) = 13,983,816 possible tickets (verified) — the real odds behind “1 in 14 million”.
  • Poker hands: C(52, 5) = 2,598,960 five-card hands — the foundation of poker odds in our Probability Calculator.
  • Pizza math: choosing 3 toppings from 12: C(12, 3) = 220 pizzas before crust and size even enter it.

Choosing Items With Repeats or Groups

The formulas above assume each item can be chosen at most once. If items may repeat (a code with repeated digits), counts follow rn-style rules instead. If a problem fixes some items in or out (“the captain must be included”), reduce n and r accordingly — include the captain, then choose the rest from the remaining pool. For data sets rather than counting, see the Standard Deviation Calculator.

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Frequently Asked Questions

How do I calculate C(10, 3)?

10! ÷ (3! × 7!) = 3,628,800 ÷ (6 × 5,040) = 120.

What is the nCr button on a calculator?

It computes combinations directly: enter n, press nCr, enter r. This page does the same with the factorial steps shown.

Can r be bigger than n?

Not without repetition — you cannot choose 5 distinct items from 3. The calculator asks for r ≤ n.

Why is C(n, r) equal to C(n, n−r)?

Choosing the r items to take is the same act as choosing the n−r items to leave behind. C(10, 3) = C(10, 7) = 120.

How many combinations are there in a 4-digit PIN?

PINs allow repeats and order matters, so it is 10⁴ = 10,000 — not a combination problem at all. The “order matters + repeats” family uses powers.

Is this combination calculator free?

Yes — free, instant, no sign-up.