Solve combinations, permutations, and draw probabilities with confidence. Compare exact, lower, and upper event chances. Understand outcomes before making probability decisions in real problems.
Use combination and permutation modes for counting tasks. Use exact probability mode for without-replacement draw problems.
| Scenario | Total Items | Selected Items | Success States | Target Successes | Interpretation |
|---|---|---|---|---|---|
| Card draw | 52 | 5 | 4 | 1 | Exactly one ace in a five-card hand |
| Quality sampling | 40 | 6 | 9 | 2 | Exactly two defectives in six tested units |
| Simple counting | 12 | 3 | — | — | Number of teams or ordered selections |
Combination: nCr = n! / (r!(n-r)!)
Permutation: nPr = n! / (n-r)!
Exact combinatorial probability: P(X=k) = [C(K,k) × C(N-K,n-k)] / C(N,n)
Here, N is total population, n is draw size, K is total success states, and k is the exact number of successes required.
It computes combinations, permutations, and exact without-replacement probabilities. This helps with cards, lottery selections, audits, inspections, and many structured counting problems.
Use combinations when order does not matter. Selecting three committee members from ten people is a combination problem because the group identity stays the same regardless of order.
Use permutations when order matters. Assigning gold, silver, and bronze positions or arranging letters into different sequences are common permutation cases.
The probability mode uses the hypergeometric model. It fits draws without replacement from a finite population containing a known number of success states.
Those values reveal how the probability was built. Probability equals favorable outcomes divided by total possible outcomes under the selected draw conditions.
Yes, for many practical cases. Extremely large factorial-based counts may display in scientific notation to keep the result readable and stable.
Cumulative lower gives the probability of getting at most k successes. Cumulative upper gives the probability of getting at least k successes.
Validation appears when values break combinatorial rules, such as choosing more items than available or entering success states larger than the population.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.