Combinations

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The number of ways to choose k items from n without regard to order.

The number of combinations C(n, k) (also written nCk or 'n choose k') counts the unordered selections of k items from a set of n distinct items. It equals the binomial coefficient n! / (k! · (n − k)!). Combinations differ from permutations (where order matters): C(n, k) = P(n, k) / k!. Symmetry: C(n, k) = C(n, n − k), so choosing k to include is equivalent to choosing n − k to exclude.

C(n, k) = n! / (k! · (n − k)!)

• C(5, 2) = 10 (ten ways to pick 2 from 5, such as {1,2}, {1,3}, ..., {4,5})
• C(10, 3) = 120
• C(52, 5) = 2,598,960 (five-card poker hands)

How to recognize it

• The problem asks 'how many ways' without regard to order
• Keywords: 'choose', 'select', 'committee', 'team', 'hand of cards'
• If rearranging the chosen items yields the same selection, use combinations, not permutations

Common mistakes

• Using permutations when order does not matter
• Forgetting the k! in the denominator
• Confusing C(n, k) with n · k or with n^k

Practice guidance

Numerical Reasoning