Question

an election ballot asks voters to select five city commissioners from a group of fifteen candidates. in how many ways can this be done?

five city commissioners can be selected from a group of fifteen candidates in \\(\square\\) different ways.

Explanation:

Step1: Identify the combination formula

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to be chosen. Here, \(n = 15\) and \(r=5\).

Step2: Calculate factorial values

First, calculate \(n!=15! = 15\times14\times13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1\), \(r!=5! = 5\times4\times3\times2\times1\), and \((n - r)!=(15 - 5)!=10! = 10\times9\times8\times7\times6\times5\times4\times3\times2\times1\).
Then, \(C(15,5)=\frac{15!}{5!(15 - 5)!}=\frac{15\times14\times13\times12\times11\times10!}{5!\times10!}\).
Cancel out the \(10!\) terms.

Step3: Simplify the expression

\(C(15,5)=\frac{15\times14\times13\times12\times11}{5\times4\times3\times2\times1}\).
\(15\div5 = 3\), \(14\div2=7\), \(12\div4 = 3\).
The expression becomes \(3\times7\times13\times3\times11\).
\(3\times7 = 21\), \(21\times13=273\), \(273\times3 = 819\), \(819\times11=45045\div1 = 3003\).

Answer:

\(3003\)