Question

1. a club is hosting an election for the positions of president, vice president, secretary, and treasurer. there are 6 candidates—student a, student b, student c, student d, student e, and student f. after voting, the first - place candidate will be president, the second - place candidate will be vice president, the third - place candidate will be secretary, and the fourth - place candidate will be treasurer. how many possible outcomes are there?

Explanation:

Step1: Identify the problem type

This is a permutation problem where we need to find the number of ways to arrange 6 candidates into 4 distinct positions (President, Vice - President, Secretary, Treasurer). The formula for permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 6\) (total number of candidates) and \(r=4\) (number of positions).

Step2: Apply the permutation formula

First, recall the factorial formula \(n!=n\times(n - 1)\times(n - 2)\times\cdots\times1\).
For \(P(6,4)=\frac{6!}{(6 - 4)!}=\frac{6!}{2!}\)
We know that \(6!=6\times5\times4\times3\times2\times1 = 720\) and \(2!=2\times1=2\)
So, \(\frac{6!}{2!}=\frac{720}{2}=6\times5\times4\times3\) (because \(\frac{6!}{2!}=\frac{6\times5\times4\times3\times2!}{2!}=6\times5\times4\times3\))
Calculate \(6\times5\times4\times3 = 360\)

Answer:

360