Combinations and probability
How many ways are there to pick 6 numbers out of 49? What is the chance of 3 successes in 10 trials? Enter the numbers for an exact answer and a chart.
Result
- Permutations (order matters)
- 10 068 347 520
- With repetition, order matters nᵏ
- 13 841 287 201
- Combinations with repetition
- 25 827 165
- 49!
- 6.082818 × 10⁶² (63 digits)
- Chance of one particular combination
- 1 : 13 983 816
C(n, k) = n! / (k! · (n − k)!)
P(n, k) = n! / (n − k)!
Combinations and permutations
Combinations count selections where order doesn’t matter: a 6-from-49 lottery has C(49, 6) = 13,983,816 possible tickets. Permutations count arrangements where order matters, such as a podium from 10 athletes: 10 · 9 · 8 = 720. Results are exact even for very large numbers.
The binomial distribution
Repeat n independent trials with the same success probability p, and the number of successes follows the binomial distribution: P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ. For example, the chance of exactly 3 heads in 10 coin tosses is about 11.7%.
The normal distribution
In a normal distribution about 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three. The tool works out the exact area under the curve for any interval.
Common questions
What is the difference between combinations and permutations?
Combinations only care what is chosen ({1, 2} = {2, 1}). Permutations also care about order (1, 2 ≠ 2, 1).
How do I enter p = 1/6?
Just type 1/6. The box accepts fractions and simple expressions.