Choosing without order: combinations

From permutations to combinations

A permutation counts a selection once for every order it can be listed in. A combination ignores order, so we divide the ordered count by the number of ways to reorder the k chosen items, which is k!.

For a five-card selection: 311,875,200 ordered deals ÷ 120 orderings of each set = 2,598,960 distinct sets.

Useful properties

These identities save time and are good checks on your arithmetic.

• Symmetry: C(n, k) = C(n, n − k). Choosing 5 cards to keep is the same as choosing 47 to leave out.
• Pascal's rule: C(n, k) = C(n − 1, k − 1) + C(n − 1, k). Either a particular card is in the selection or it is not.
• Total subsets: C(n, 0) + C(n, 1) + … + C(n, n) = 2^n.

Counting a pattern: exactly one pair

Counting a category is a recipe of independent choices, each handled by the multiplication principle. To count five-card selections with exactly one pair (two cards of the same rank and three cards of three other, different ranks):

• Choose the rank of the pair: 13 ways.
• Choose which 2 of that rank's 4 suits: C(4, 2) = 6 ways.
• Choose 3 other distinct ranks: C(12, 3) = 220 ways.
• Choose a suit for each of those three cards: 4 × 4 × 4 = 64 ways.

Individual outcomes versus events

Every specific five-card set is equally likely. A pattern such as “exactly one pair” is an event that gathers many sets together. Rare events are rare not because their members are unusual, but because there are few members. Keeping this distinction clear prevents many intuitions about randomness from going wrong.

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