What is a Probability & Combinatorics Calculator?
This calculator computes permutations, combinations, and probabilities with formulas shown.
How it's calculated
Permutation nPr = n! ÷ (n−r)! (order matters)
Combination nCr = n! ÷ (r!(n−r)!) (order doesn't matter)
Step-by-step example
How many ways to choose 2 representatives from 5 people (combination)?
- Step 1 Apply the formula: 5C2 = 5! ÷ (2!×3!)
- Step 2 Calculate: (5×4) ÷ (2×1) = 20÷2
- Step 3 Result: 10 ways
Interpretation: If order mattered (permutation, 5P2), the answer would double to 20 ways.
Permutations vs. combinations — what's the difference?
The only question is whether order matters. Electing a president and vice president is a permutation (President A/VP B is different from President B/VP A); selecting two committee representatives is a combination (A,B is the same group as B,A). Combinations are smaller than permutations because they divide out the r! ways of ordering the same selected group.
Good to know
- The odds of winning a typical 6-of-49 lottery are about 1 in 14 million — a straightforward combination calculation (49 choose 6), often compared to the odds of being struck by lightning.
- The birthday paradox is famous: with just 23 people in a room, there's over a 50% chance two share a birthday — counterintuitive because it's not 'my birthday matching yours,' but any pair among 253 possible pairs matching.
- The Monty Hall problem also defies intuition: after picking one of three doors and having a losing door revealed, switching doubles your odds of winning (from 1/3 to 2/3) — a result that famously stumped many mathematicians when first published.
- 0! = 1 by definition — it makes sense once you realize 'arranging nothing' has exactly one way to do it (doing nothing), and it keeps the combinatorial formulas consistent.
- Probability theory has roots in gambling — the correspondence between Pascal and Fermat in the 17th century, prompted by a gambler's question about dividing stakes in an interrupted game, is considered the field's founding moment.
Frequently asked questions
Q. Should I use permutations or combinations?
If order affects the outcome, use permutations. If it doesn't, use combinations. 'Select then arrange' is permutation; 'just select' is combination.
Q. What if repetition is allowed?
Permutations with repetition use n^r; combinations with repetition use (n+r−1)C(r).
Q. If probability is 0.5, does trying twice guarantee success?
No. Both attempts failing has a 25% chance — each trial is independent.