Chapter 13.9, Problem 1AGP

To determine

The number of ways to arrange the first and second place in a Battle of bands is to be described.

Answer to Problem 1AGP

Through the $56$ ways first and second place in a Battle of bands can arranged.

Explanation of Solution

Given:

  $\begin{array}{l}n=8\\ r=\left\{1,2\right\}=2\end{array}$

Concept Used:

The formula for combination is given by the following expression:

  $C{}_r=\frac{n!}{\left(n-r\right)!}$

Calculation:

The number of ways to arrange the first and second place in a Battle of bands is calculated by using the combination:

  $\begin{array}{l}C{}_r=\frac{n!}{\left(n-r\right)!}\\ C{}_2=\frac{n!}{\left(n-r\right)!}\\ C{}_2=\frac{8!}{\left(8-2\right)!}\\ C{}_2=\frac{8!}{6!}=\frac{\left(8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\right)}{\left(6\times 5\times 4\times 3\times 2\times 1\right)}\\ C{}_2=8\times 7\\ C{}_2=56\end{array}$

Conclusion:

Hence, the number of ways to arrange the first and second place in a Battle of bands is $C{}_2=56$