Combinations Calculator

Compute combinations \( C(n, k) \) and combinations with repetition effortlessly.

Combinations Solver

Enter the total items (\( n \)) and items to choose (\( k \)) to calculate combinations.

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About Combinations

Combinations count the number of ways to select \( k \) items from \( n \) items where order does not matter. The standard formula is:

  • \( C(n, k) = \frac{n!}{k! (n - k)!} \)
  • For repetition: \( C(n + k - 1, k) = \binom{n + k - 1}{k} \)

Examples:

  • Team Selection: How many ways to choose 3 players from 5? \( C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{120}{6 \cdot 2} = 10 \). Answer: 10 ways.
  • Ice Cream Flavors: How many ways to choose 2 scoops from 3 flavors (repetition allowed)? \( C(3 + 2 - 1, 2) = C(4, 2) = \frac{4!}{2!2!} = 6 \). Answer: 6 ways (e.g., AA, AB, AC, BB, BC, CC).
  • Committee Formation: How many ways to select 2 members from 4 people? \( C(4, 2) = \frac{4!}{2!2!} = \frac{24}{4} = 6 \). Answer: 6 ways.