# Permutations and Combinations

Another common confusion is between permutations and combinations. A permutation is a set of items whose order is specified. A combination is a set of items whose order is not specified. Imagine, for example, that three cards—the jack, queen, and king of spades—are facedown in front of you. If you pick two of these cards in turn, there are three possible combinations: jack and queen, jack and king, and queen and king. In contrast, there are six possible permutations: jack then queen, queen then jack, jack then king, king then jack, queen then king, and king then queen.

Rule 2 is used to calculate probabilities of permutations—of conjunctions of events in a particular order. For example, if you flip a fair coin twice, what is the probability of its coming up heads and tails in that order (that is, heads on the first flip and tails on the second flip)? Since the flips are independent, Rule 2 tells us that the answer is 1/2 × 1/2 = 1/4. This answer is easily con- firmed by counting the possible permutations (heads then heads, heads then tails, tails then heads, tails then tails). Only one of these four permutations (heads then tails) is a favorable outcome.

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We need to calculate probabilities of combinations in a different way. For example, if you flip a fair coin twice, what is the probability of its landing heads and tails in any order? There are two ways for this to happen. The coin could come up either heads then tails or tails then heads.