Question

at a benefit concert, seven bands have volunteered to perform but there is only enough time for five of the bands to play. how many lineups are possible?
(simplify your answer. type an integer or a fraction.)

Explanation:

Step1: Identify the permutation formula

The number of permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\). Here \(n = 7\) (total number of bands) and \(r=5\) (number of bands that can play).

Step2: Calculate the factorial values

We know that \(n!=n\times(n - 1)\times\cdots\times1\). So \(7! = 7\times6\times5\times4\times3\times2\times1=5040\) and \((7 - 5)!=2!=2\times1 = 2\).

Step3: Substitute into the formula

\(P(7,5)=\frac{7!}{(7 - 5)!}=\frac{7!}{2!}=\frac{5040}{2}\).

Step4: Simplify the expression

\(\frac{5040}{2}=2520\).

Answer:

\(2520\)