Question

an election ballot asks voters to select four city commissioners from a group of fifteen candidates. in how many ways can this be done? four city commissioners can be selected from a group of fifteen candidates in \\(\square\\) different ways.

Explanation:

Step1: Identify the problem type

This is a combination problem since the order of selecting the commissioners does not matter. The formula for combinations is \( C(n, r)=\frac{n!}{r!(n - r)!} \), where \( n = 15 \) (total candidates) and \( r = 4 \) (commissioners to select).

Step2: Calculate factorials

First, find \( n! = 15! = 15\times14\times13\times12\times11\times\cdots\times1 \), \( r! = 4! = 4\times3\times2\times1 = 24 \), and \( (n - r)! = (15 - 4)! = 11! = 11\times10\times\cdots\times1 \).

Step3: Simplify the combination formula

\( C(15, 4)=\frac{15!}{4!(15 - 4)!}=\frac{15\times14\times13\times12\times11!}{4!\times11!} \). The \( 11! \) terms cancel out, leaving \( \frac{15\times14\times13\times12}{4\times3\times2\times1} \).

Step4: Compute the numerator and denominator

Numerator: \( 15\times14 = 210 \), \( 210\times13 = 2730 \), \( 2730\times12 = 32760 \). Denominator: \( 4\times3 = 12 \), \( 12\times2 = 24 \), \( 24\times1 = 24 \). Then, \( \frac{32760}{24}=1365 \).

Answer:

1365