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A combination counts the number of ways to choose r items from a set of n items when order does not matter. The notation C(n, r), nCr, or "n choose r" all refer to the same calculation.
Formula: C(n, r) = n! / (r! × (n − r)!). For example, how many 5-card hands can be dealt from a 52-card deck? C(52, 5) = 52! / (5! × 47!) = 2,598,960 possible hands.
Combinations appear everywhere: lottery draws, committee selection, quality control sampling, probability calculations, and the binomial theorem. The key distinguishing feature: swapping two chosen items does NOT create a new combination (unlike permutations).
Worked example — a pizza shop offers 12 toppings; you can choose any 3. How many distinct pizzas are possible? C(12, 3) = 12! / (3! × 9!) = (12 × 11 × 10) / (3 × 2 × 1) = 220 combinations.
C(n,r) = n! / (r! × (n − r)!)
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This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.