Question

1. how many ways can a president and a vice-president be chosen (order matters)?

Explanation:

To solve the problem of finding the number of ways to choose a president and a vice - president (where order matters), we assume there are 8 people to choose from (since the answer is 56, and we know that for permutations \(P(n,r)=\frac{n!}{(n - r)!}\), and if \(r = 2\), we can work backwards. Let's assume the number of people is \(n\), and we want to find \(n\) such that \(P(n,2)=n\times(n - 1)=56\). Solving \(n^{2}-n - 56=0\), factoring gives \((n - 8)(n+7)=0\), so \(n = 8\) as the number of people can't be negative).

Step 1: Recall the permutation formula

The number of permutations of \(n\) objects taken \(r\) at a time is given by the formula \(P(n,r)=\frac{n!}{(n - r)!}\). When \(r = 2\) (choosing a president and a vice - president, 2 positions), the formula simplifies to \(P(n,2)=n\times(n - 1)\) because \(\frac{n!}{(n - 2)!}=\frac{n\times(n - 1)\times(n - 2)!}{(n - 2)!}=n\times(n - 1)\).

Step 2: Determine the value of \(n\)

We know that the result of \(P(n,2)\) is one of the options. Let's assume there are \(n\) people. We test the formula with the idea that if we have \(n\) people, the number of ways to choose a president ( \(n\) choices) and then a vice - president ( \(n - 1\) choices, since one person is already chosen as president) is \(n\times(n - 1)\).

We check the options:

• If we assume \(n = 8\), then \(n\times(n - 1)=8\times7 = 56\).

Answer:

The number of ways to choose a president and a vice - president (order matters) is \(\boldsymbol{56}\) (assuming there are 8 people to choose from).