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C Combination Calculator — C(n, r)

Calculate the number of ways to choose r items from n items when order does not matter. C(n, r) = n! / (r! × (n − r)!). Includes Pascal's triangle, step-by-step working, and symmetry property.

Total number of items to choose from

Number of items to select (order ignored)

Combination Formula

C(n, r) = n! / ( r! × (n − r)! )
C(n, r) = P(n, r) / r!
C(n, r) = C(n, n − r) ← symmetry

A combination counts selections where order does not matter. {A, B, C} and {C, B, A} are the same combination. The denominator r! divides out all orderings of the chosen items.

How to Calculate C(10, 3)

  1. 1
    Write the formula
    C(10, 3) = 10! / (3! × (10−3)!) = 10! / (3! × 7!)
  2. 2
    Simplify the numerator
    Cancel the 7! from top and bottom: 10 × 9 × 8 / 3!
  3. 3
    Compute denominator
    3! = 3 × 2 × 1 = 6
  4. 4
    Divide
    (10 × 9 × 8) / 6 = 720 / 6 = 120. There are 120 ways to choose 3 from 10.

Frequently Asked Questions

A combination is a selection of r items from a set of n items where order does not matter. For example, choosing 3 toppings for a pizza from a list of 10 is a combination problem — the order you pick them makes no difference to the final pizza.

Use a combination when the group composition is what matters (selecting a committee, choosing lottery numbers). Use a permutation when the order or arrangement matters (assigning first/second/third place, creating a password). P(n,r) = C(n,r) × r!.

Pascal's triangle is a triangular array where each number equals the sum of the two numbers directly above it. Row n gives the binomial coefficients C(n,0), C(n,1), ..., C(n,n). It is used in combinatorics, probability, and algebra (binomial theorem).

The UK Lotto draws 6 balls from 59. The number of possible tickets is C(59, 6) = 45,057,474. For the classic 6/49 lottery: C(49, 6) = 13,983,816. Your chance of winning with one ticket is roughly 1 in 14 million.

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