Permutations and combinations are the heart of combinatorial mathematics.
About this calculator
Permutations and combinations are the heart of combinatorial mathematics. Whenever you need to count 'how many ways can this happen', one of these tools will be the answer. The distinction is simple but important: permutations care about order, combinations don't. Picking 3 winners for gold, silver and bronze medals from 10 athletes is permutations -- swapping who gets gold and silver matters. Picking 3 friends to invite to a movie from 10 candidates is combinations -- the order you list them doesn't change the group.
This calculator handles both, plus the related factorial calculations, plus the 'with repetition' versions which come up often in passwords, license plates and dice rolls. The formula links combinations and permutations elegantly: C(n,r) = P(n,r) / r! -- because each combination of r items can be re-ordered in r! different ways, all counted as permutations. So combinations are always smaller (or equal when r = 0 or r = n).
Real-world applications are everywhere. Lottery odds use combinations -- the order of numbers doesn't matter. Lock combinations actually need permutations -- order matters despite the name. Card games use a mix: poker hands are combinations (order of cards in your hand doesn't matter), but the number of ways to deal a sequence of cards is permutations. In statistics, the binomial coefficient C(n,k) is the same as the combination formula and counts the number of ways an event can happen k times in n trials. To put these counts to use in probability calculations, head to the [[probability]] calculator. For algebraic equations the [[quadratic]] calculator helps, and geometry calculators ([[cone]], [[cylinder]], [[sphere]]) often involve volume and surface formulas with similar polynomial structure.
Common uses
- Class 11/12 Mathematics permutation and combination problems
- Probability calculations for poker, lottery and card games
- Counting distinct passwords or PIN possibilities for security analysis
- Tournament seeding and team selection counts
- Inventory and combinatorial design in operations research
Frequently asked questions
When do I use permutations vs combinations?
Use permutations when order matters -- ranking, podium positions, sequence of letters, racing positions. Use combinations when order doesn
What is 0!?
By definition, 0! = 1. This makes formulas like C(n,n) = n!/(n! × 0!) = 1 work correctly -- there
Why does the calculator show 'too large to display' for big n?
Factorials grow extremely fast. 20! is already 2.4 × 10^18, and 170! is the largest factorial JavaScript can represent. Beyond n = 170 the answer overflows, so we show scientific notation or
What does 'with repetition' mean?
With repetition allowed, the same item can be picked multiple times. Choosing a 4-digit PIN allows the same digit to repeat, so it
How does this connect to Pascal's triangle?
Pascal