Question

the members of the student activity council on your campus are meeting to select three speakers for a month - long event celebrating artists and entertainers. the first names of the choices are ben, wil, stewart, and hilary. how many different ways can the three speakers be selected?
the three speakers can be selected \square different ways.

Explanation:

Step1: Identify the problem type

This is a combination problem since we are selecting 3 speakers from 4 people (Ben, Wil, Stewart, Hilary) and the order of selection doesn't matter (as we are just choosing a group of 3). The formula for combinations is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose. Here, \( n = 4 \) and \( k=3 \).

Step2: Calculate the combination

First, find the factorial values. The factorial of a number \( n \) (denoted as \( n! \)) is the product of all positive integers up to \( n \). So, \( 4! = 4\times3\times2\times1=24 \), \( 3! = 3\times2\times1 = 6 \), and \( (4 - 3)! = 1! = 1 \).

Now, substitute these values into the combination formula:

$$ C(4, 3)=\frac{4!}{3!(4 - 3)!}=\frac{24}{6\times1}=\frac{24}{6} = 4 $$

Alternatively, we can think of it as the number of ways to leave out 1 person from 4, and there are 4 ways to leave out 1 person (leave out Ben, leave out Wil, leave out Stewart, leave out Hilary), so the number of ways to choose 3 is 4.

Answer:

4