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Home » Math & Conversions » Algebra » Combination Combination Calculator Find the number of ways to choose $k$ items from a set of $n$. Our tool also calculates **Permutations** and handles both **Repetition** and **No Repetition** scenarios. Total Items (n): Items Chosen (k): Calculate Combinations & Permutations Combinations (No Repetition) C(n, k) - Permutations (No Repetition) P(n, k) - Combinations (With Repetition) - Permutations (With Repetition) - Combinations vs. Permutations: When Does Order Matter? The first decision in combinatorics is determining if you are dealing with a combination or a permutation. Decision Matrix: Scenario Order Matters? Repetition Allowed? Formula Type Selecting a podium finish (1st, 2nd, 3rd) Yes (Permutation) No Permutation P(n, k) Selecting 3 friends for a party No (Combination) No Combination C(n, k) Locker code (digits can repeat) Yes (Permutation) Yes nᵏ Choosing donuts (multiple of the same type) No (Combination) Yes C(n + k - 1, k) Key Formulas (No Repetition) Permutation: $$ P(n, k) = \frac{n!}{(n - k)!} $$ Combination: $$ C(n, k) = \frac{n!}{k!(n - k)!} $$ Frequently Asked Questions (FAQ) What is the formula for a combination without repetition? The formula is: $C(n, k) = \frac{n!}{k!(n - k)!}$. This calculates the number of ways to choose $k$ items from a set of $n$ items, where the order of selection does not matter. When is a result a Permutation? A result is a permutation when the **order of the items matters**. For example, forming the word 'CAT' is different from forming 'ACT', even though they use the same letters. If you are picking a password, a winning race order, or assigning roles, use permutations. Can combinations have repetition? Yes. Combinations with repetition are used when the items being chosen can be selected more than once (e.g., choosing 3 cookies from 5 available flavors). The formula for combinations with repetition is $C(n + k - 1, k)$. Related Calculators Permutation Calculator Probability Calculator Factorial Calculator Mode Calculator Roots Calculator Logarithm Calculator Exponent Calculator
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