Combination Calculator
Use this free Combination Calculator to quickly get accurate results in seconds, free and instantly offline with no signup required ever, trusted by professionals in Austin, Chicago, Miami, Denver, Seattle, and New York for reliable daily use without accounts, subscriptions, limits, or delays on any device for everyday personal and business workflows daily.
Calculate combinations C(n,r).
Calculator Inputs
About Combinations
Use this free Combination Calculator to quickly get accurate results in seconds, free and instantly offline with no signup required ever, trusted by professionals in Austin, Chicago, Miami, Denver, Seattle, and New York for reliable daily use without accounts, subscriptions, limits, or delays on any device for everyday personal and business workflows daily.
How counting works in seconds: Enter n items with r selections to get combinations (order-free) plus permutations (ordered) together. Results appear at $0 cost in your browser.
Why counting matters for USA odds on 09/22/2026: Intuition fails factorials — 6-from-49 lotteries hide 14 million outcomes. Austin players checking $20M jackpot odds face true probabilities. Chicago poker rooms raking $900,000 tournaments set structures by combinatorics.
Fantasy drafts: Miami leagues drafting $500,000 seasons compute pick values from position scarcity. Denver hiring panels shortlisting $300,000 searches interview slates fairly.
Test matrices: Seattle QA sampling $2M configuration matrices cover pairs without exhaustive runs. New York insurers pricing $90M risk pools model claim combinations.
Order discipline: Committees use combinations; passwords use permutations — swapping them inflates counts astronomically.
Corral seeding: Boston marathons seeding $300,000 runner corrals assign waves by qualifying times. Chicago bars running $150,000 poker leagues structure payouts by entrant counts on 09/22/2026.
Boat raffles: Miami boat shows raffling $2M center consoles disclose odds computed combinatorially. Phoenix radio stations awarding $400,000 prize trucks file contest rules.
Jury pools: Denver courts pooling civic jurors randomize panels without replacement bias. Every qualified resident deserves equal odds.
Intuition repair: Birthday paradoxes prove humans misjudge coincidences — trust factorials over feelings in every estimate.
Large-n tools: Factorials overflow fast — use logarithmic or iterative computation beyond twenty items.
Marathon waves: Boston marathons seeding $300,000 runner corrals assign waves by qualifying times. Chicago bars running $150,000 poker leagues structure payouts by entrant counts on 09/22/2026.
Boat raffles: Miami boat shows raffling $2M center consoles disclose odds computed combinatorially. Phoenix radio stations awarding $400,000 prize trucks file contest rules.
Civic panels: Denver courts pooling civic jurors randomize panels without replacement bias. Every qualified resident deserves equal odds always.
Overflow guards: Factorials overflow fast beyond twenty items — use logarithmic or iterative computation for large-n sanity.
Where it is used
Austin lottery players check $20M jackpot odds, Chicago poker rooms rake $900,000 tournaments, Miami fantasy leagues draft $500,000 seasons, Denver hiring panels shortlist $300,000 searches, Seattle QA samples $2M test matrices, New York insurers price $90M risk pools, Boston marathons seed $300,000 corrals, Chicago bars run $150,000 poker leagues, Miami boat shows raffle $2M center consoles, Denver juries pool civic panels, and Phoenix radio giveaways award $400,000 prize trucks.
How to use the Combinations
- Total Items (n) — enter the value (e.g. 10).
- Select Items (r) — enter the value (e.g. 3).
- Calculate — press the Calculate button to see the result instantly above.
Formula
P(n,r) = n! / (n-r)! C(n,r) = n! / (r! * (n-r)!)
Examples
Lottery odds
Poker hands
Team picks
Frequently Asked Questions
- When should I use permutations versus combinations for combination?
- Use permutations when the order of selection matters, such as arranging items, forming passwords, or ranking candidates. Use combinations when order does not matter, such as selecting a committee, choosing lottery numbers, or picking items from a menu. The key question is whether rearranging the same items creates a different combination outcome.
- Why are permutations always greater than or equal to combinations for combination?
- Permutations count every possible arrangement of the selected items, while combinations count each unique group only once. Since each combination of r items can be arranged in r factorial different ways, permutations equal combinations multiplied by r factorial. This relationship is fundamental to combination counting problems.
- What is the difference between permutations with and without repetition for combination?
- Without repetition, each item can be selected only once. With repetition, items can be reused. For example, a 4-digit PIN allows repetition, while selecting 4 cards from a deck does not. The Combination Calculator computes permutations and combinations without repetition for accurate combination results.
- How are permutations and combinations used in combination probability?
- To find the probability of a specific outcome, divide the number of favorable arrangements by the total number of possible arrangements. This approach solves problems ranging from card games to quality control, making combination probability calculations systematic and accurate.
- Can I use the Combination Calculator for lottery odds calculations?
- Yes, lottery odds are calculated using combinations. For example, choosing 6 numbers from 49 gives over 13.9 million possible combinations, meaning the odds of matching all six are approximately 1 in 14 million. Use the combinations result to understand your chances in any combination lottery or random selection scenario.