Question

this exercise can be solved using permutations even though the problem statement will not always include a form of the word \permutation\ or \arrangement\ or \ordering.\ how many ways can a president, vice - president, secretary, and treasurer be chosen from a committee of 8 people? the number of ways to choose a president, vice - president, secretary, and treasurer is \boxed{}

Explanation:

Step1: Identify the problem type

We need to find the number of ways to choose 4 distinct positions (president, vice - president, secretary, treasurer) from 8 people. This is a permutation problem because the order of selection matters (a person in president position is different from the same person in vice - president position). The formula for permutations is \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to be selected. Here, \(n = 8\) and \(r=4\).

Step2: Apply the permutation formula

First, calculate \(n!=8! = 8\times7\times6\times5\times4\times3\times2\times1\) and \((n - r)!=(8 - 4)!=4!=4\times3\times2\times1\).
Then \(P(8,4)=\frac{8!}{(8 - 4)!}=\frac{8!}{4!}=\frac{8\times7\times6\times5\times4!}{4!}\).
The \(4!\) terms in the numerator and denominator cancel out, leaving us with \(8\times7\times6\times5\).

Step3: Calculate the result

\(8\times7 = 56\), \(56\times6=336\), \(336\times5 = 1680\).

Answer:

1680