Chapter SH, Problem 12.6MPS

a.

To determine

To find the number of ways that the band director can assign the first chair, second chair and third chair.

Answer to Problem 12.6MPS

  $2016$ .

Explanation of Solution

Given:

There are $6$ girls and $4$ boys in a high school band.

Calculation:

The following seating arrangements can be possible for assign the first chair, second chair and third chair to $6$ girls and $4$ boys in a high school band.

  $GGG,GGB,BGG,GBG,BBG,BGB,GBB,BBB$

Now adding the corresponding number of ways

  $\begin{array}{l}GGG,GGB,BGG,GBG,BBG,BGB,GBB,BBB\\ =3!(6\times 5\times 4)+3\left(2!(6\times 5\times 4\right))+3\left(2!(4\times 3\times 6\right)+3!(4\times 3\times 2)\\ =720+720+432+144\\ =2016\end{array}$

b.

To determine

To find the probability that the first three chairs will be assigned to boys.

Answer to Problem 12.6MPS

  $\frac{1}{14}$

Explanation of Solution

Given:

There are $6$ girls and $4$ boys in a high school band.

Calculation:

The number of ways to assign the first three chairs to $4$ boys are

  $3!(4\times 3\times 2)=144$

Therefore, the probability is $\frac{144}{2016}=\frac{1}{14}$