How to use it
- Choose combinations, permutations, arrangements, binomial or simple probability.
- Enter n and r, the group sizes, or the success probability.
- Tick With repetition if items can repeat.
- Read the exact count or the probability.
Combinations vs. permutations
The key question is whether order matters.
- Combinations (nCr): order does not matter. Choosing 5 lottery numbers or a 5-card poker hand.
- Permutations (nPr): order matters. Awarding gold, silver and bronze, or setting a 4-digit PIN.
| Formula | Order matters? | Repetition? |
|---|---|---|
| C(n, r) = n! / (r! (n − r)!) | No | No |
| C(n + r − 1, r) | No | Yes |
| P(n, r) = n! / (n − r)! | Yes | No |
| n^r | Yes | Yes |
Results are exact even when they run to hundreds of digits, because the calculator uses big-integer math in your browser.
Examples
- Poker hands: C(52, 5) = 2,598,960 possible 5-card hands.
- Powerball: choosing 5 white balls from 69 gives C(69, 5) = 11,238,513 combinations. Multiply by the 26 possible red Powerballs and the jackpot odds are 1 in 292,201,338.
- Medals: 10 runners, gold, silver and bronze: P(10, 3) = 10 × 9 × 8 = 720 ways.
- PIN codes: 4 digits with repetition: 10⁴ = 10,000.
- Ice cream: 3 scoops from 8 flavors, repeats allowed, order ignored: C(10, 3) = 120.
Arrangements with repeated items
To count the distinct orders of items when some are identical, divide n! by the factorial of each group size.
Example: the letters of BOOK (B, O, O, K) can be arranged in 4! / 2! = 12 ways; enter the group sizes 1, 2, 1.
MISSISSIPPI has 11 letters: M once, I four times, S four times, P twice. That gives 11! / (1! · 4! · 4! · 2!) = 34,650 arrangements.
Binomial probability
Use the binomial mode when you repeat an experiment n times, each trial has two outcomes, and the success probability p stays the same.
P(X = k) = C(n, k) · p^k · (1 − p)^(n − k)
Example: flip a fair coin 10 times. The probability of exactly 5 heads is C(10, 5) / 2¹⁰ = 252 / 1,024 ≈ 24.61%. The calculator also shows P(X ≤ k), P(X ≥ k) and the expected value n · p.
Simple probability and odds
For equally likely outcomes, probability = favorable outcomes / total outcomes. Rolling a 6 on a die: 1/6 ≈ 16.67%. The complement, not rolling a 6, is 5/6 ≈ 83.33%.
Odds compare favorable to unfavorable outcomes: 1 : 5 for rolling a 6. Odds and probability are not the same thing, which is a common source of confusion; betting odds also include the bookmaker's margin.
For factorials on their own, use the factorial calculator.
Common mistakes
- Using permutations when order does not matter. That overcounts by a factor of r!. A 3-person committee from 10 people is C(10, 3) = 120, not P(10, 3) = 720.
- Adding probabilities of events that can happen together. P(A or B) = P(A) + P(B) − P(A and B).
- The gambler's fallacy. After five heads in a row, the next fair coin flip is still 50% heads; independent trials have no memory.
- Confusing odds with probability. Odds of 1 : 5 mean a probability of 1/6, not 1/5.
- Forgetting the complement. The chance of at least one six in four rolls is easiest as 1 − (5/6)⁴ ≈ 51.8%.
Frequently asked questions
What is the difference between a combination and a permutation?
In a combination, order does not matter; in a permutation it does. Choosing 3 people for a committee is a combination; electing a president, VP and secretary is a permutation.
How do I calculate nCr?
n! / (r! × (n − r)!). For C(10, 3): 3,628,800 / (6 × 5,040) = 120.
What are the odds of winning the Powerball jackpot?
1 in 292,201,338: C(69, 5) white-ball combinations times 26 Powerball numbers.
What is binomial probability?
The probability of exactly k successes in n independent trials with the same success probability p.
Can the results be very large?
Yes. The calculator uses exact big-integer arithmetic and also shows scientific notation for huge counts.
Last updated: November 12, 2026