A club with 15 members is to choose four officers–president, vice president, secretary, and treasurer. In how many ways can these offices be filled?

검증된 단계별 안내

Step 1: Recognize that this is a permutation problem because the order in which the officers are chosen matters (e.g., president is different from vice president).

Step 2: Use the formula for permutations, which is P(n, r) = n! / (n - r)!, where n is the total number of items (15 members) and r is the number of items to choose (4 officers).

Step 3: Substitute the values into the formula: P(15, 4) = 15! / (15 - 4)!.

Step 4: Simplify the denominator: (15 - 4)! = 11!, so the formula becomes P(15, 4) = 15! / 11!.

Step 5: Cancel out the common factorial terms (11!) in the numerator and denominator, leaving P(15, 4) = 15 × 14 × 13 × 12. Multiply these terms to find the total number of ways the offices can be filled.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Permutations

Permutations refer to the different arrangements of a set of items where the order matters. In this scenario, the positions of president, vice president, secretary, and treasurer are distinct, meaning that the arrangement of members in these roles is crucial. The formula for permutations is n! / (n - r)!, where n is the total number of items, and r is the number of items to arrange.

Factorial

A factorial, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics used to calculate permutations and combinations. For example, 5! equals 5 × 4 × 3 × 2 × 1 = 120. Understanding factorials is essential for determining the number of ways to arrange members in the club's officer positions.

Combinatorial Counting

Combinatorial counting involves calculating the number of ways to select or arrange items based on specific criteria. In this question, we are interested in how to fill four distinct officer roles from a group of 15 members. This requires applying the principles of permutations, as the order of selection is important, leading to a systematic approach to counting the arrangements.