Question

a club with seventeen members is to choose three officers: president, vice - president, and secretary - treasurer. if each office is to be held by one person and no person can hold more than one office, in how many ways can these offices be filled?
ways

Explanation:

Step1: Use permutation formula

The number of permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\). Here \(n = 17\) (total number of members) and \(r=3\) (number of offices).

Step2: Calculate factorial values

\(P(17,3)=\frac{17!}{(17 - 3)!}=\frac{17!}{14!}\). Since \(n!=n\times(n - 1)\times\cdots\times(n-r+1)\times(n - r)!\), then \(\frac{17!}{14!}=17\times16\times15\).

Step3: Compute the product

\(17\times16\times15=(17\times16)\times15 = 272\times15=4080\).

Answer:

\(4080\)