How can you tell whether a counting problem is a permutation or a combination problem? Give examples to support your answer.
One way to distinguish between a permutation and combination problem is to consider whether order matters in the given scenario.
Permutation:
- Order matters in permutation problems, meaning that the arrangement of the elements being counted is important.
- For example, if we are trying to find the number of ways to arrange three students (A, B, C) in a line, we would use permutations. The possible arrangements would be ABC, ACB, BAC, BCA, CAB, CBA, for a total of 6 permutations.
- Another example would be if we are selecting a president, vice president, and secretary from a group of 10 people. The order in which the positions are filled matters, so we would use permutations to calculate the number of possible outcomes.
Combination:
- Order does not matter in combination problems, meaning that the arrangement of the elements being counted is irrelevant.
- For example, if we are selecting a group of 2 people from a pool of 5 candidates, to form a committee, the order in which the people are selected does not matter. The combination of selecting 2 people out of 5 would be denoted as "5 choose 2" or 5C2 and calculated as 10 possible combinations.
- Another example would be selecting a subset of fruits from a basket containing apples, oranges, and bananas. If we are asked to choose 2 fruits for a snack, the order in which the fruits are selected does not matter, so we would use combinations to find the number of possible choices.