Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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When calculating combinations, what does the term n!/r!(n-r)! represent?

The number of permutations of r items
The number of arrangements of n items
The number of ways to choose r items from a set of n
The total arrangements of n items

The correct answer is: The number of ways to choose r items from a set of n

The expression n!/r!(n-r)! is specifically designed to calculate combinations, which are used when the order of selection does not matter. This formula arises from the need to determine how many different ways you can choose r items from a total of n distinct items without regard to the order of selection. In this formula, n! (n factorial) represents the total number of ways to arrange all n items. The denominator, r!(n-r)!, adjusts for the overcounting that occurs because the order of selection within the r items does not matter (hence r!) and because the arrangement of the remaining (n-r) items also does not affect the selection of the r items (hence (n-r)!). Thus, when you simplify n!/r!(n-r)!, you effectively calculate just the unique groupings of r items from the total of n. This clearly indicates the concept of choosing r items from n without concern for order, confirming that the expression represents the number of ways to choose r items from a set of n.