Question

from 7 people, how many ways to choose a president, vice - president, and secretary (all distinct)?

Explanation:

Step1: Identify the problem type

This is a permutation problem since the order of selection (president, vice - president, secretary) matters. The formula for permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 7\) (total number of people) and \(r=3\) (number of positions to fill: president, vice - president, secretary).

Step2: Apply the permutation formula

We know that \(n = 7\) and \(r=3\). So we calculate \(P(7,3)=\frac{7!}{(7 - 3)!}=\frac{7!}{4!}\). Since \(n!=n\times(n - 1)\times\cdots\times1\), we can expand \(7! = 7\times6\times5\times4!\) and \(4! = 4!\). Then \(\frac{7!}{4!}=\frac{7\times6\times5\times4!}{4!}=7\times6\times5\).

Step3: Calculate the result

\(7\times6\times5 = 210\).

Answer:

210