Probability Calculator
Calculate permutations (nPr), combinations (nCr), factorials, and binomial distribution probabilities.
Permutations & Combinations (nPr / nCr)
Combinations (\(nCr\))
120
Order does NOT matter
Permutations (\(nPr\))
720
Order matters
Factorial (\(n!\))
3628800
Total arrangements
Binomial Probability Calculator
Exact Probability \(P(X = k)\)
0.1171875
Cumulative \(P(X \le k)\)
0.171875
Cumulative \(P(X \ge k)\)
0.9453125
Expected Value / Mean (\(\mu = n \cdot p\))
5
Probability & Combinatorics Formulas
View All Math Formulas Handbook →Combinations Formula ($nCr$)
Number of ways to choose $r$ items without regard to order.
$$ \binom{n}{r} = C(n, r) = \frac{n!}{r!(n - r)!} $$
How to calculate:
$$ C(n, r) = \frac{\prod_{i=1}^{r} (n - i + 1)}{\prod_{i=1}^{r} i} $$
Permutations Formula ($nPr$)
Number of ways to arrange $r$ items in a sequence (order matters).
$$ P(n, r) = \frac{n!}{(n - r)!} $$
How to calculate:
$$ P(n, r) = n \times (n - 1) \times \dots \times (n - r + 1) $$
Binomial Distribution Formulas
Probability of exactly $k$ successes in $n$ independent Bernoulli trials with probability $p$.
$$ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} $$
Cumulative probability & Expected value:
$$ P(X \le k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1 - p)^{n - i}, \quad \mu = n \times p, \quad \sigma^2 = n p (1 - p) $$