Chapter 9.1, Problem 56E

Solution

To state: Which of the chart is more reasonable if a director casting a play with two female leads and wants to have a chance to audition the actresses two at a time to get a feeling for how well they work together. The given chart is:

  

First chart is more reasonable.

Given information:

The given chart is:

  

Explanation:

It can be seen that for three persons, the total time required is 10 minutes.

If the auditions of two actresses are taken together, the number of auditions will be combination of them.

Let’s consider the time required for one audition is: x

Then the time required for audition is:

  $\begin{array}{l}{C}_{3_2}\cdot x=10\\ 3\cdot x=10\\ x=\frac{10}{3}\\ x\approx 3.3\end{array}$

Each person’s audition will require 3.3 minutes.

If there are 6 persons the required time will become:

  $\begin{array}{c}{C}_{6_2}\cdot \left(3.3\right)=\frac{6!}{2!\left(6-2\right)!}\\ =\frac{6!}{2!4!}\left(3.3\right)\\ =49.5\end{array}$

Similarly, for 9 persons the required time will become:

  $\begin{array}{c}{C}_{9_2}\cdot \left(3.3\right)=\frac{9!}{2!\left(9-2\right)!}\\ =\frac{9!}{2!7!}\left(3.3\right)\\ =118.8\end{array}$

For 12 persons and 15 persons the required time will become:

  $\begin{array}{c}{C}_{{12}_2}\cdot \left(3.3\right)=\frac{12!}{2!\left(12-2\right)!}\\ =\frac{12!}{2!10!}\left(3.3\right)\\ =217.8\\ C_{{15}_2}\cdot \left(3.3\right)=\frac{15!}{2!\left(15-2\right)!}\\ =\frac{15!}{2!13!}\left(3.3\right)\\ =364.5\end{array}$

From the above calculation it is clear that the first chart is more reasonable.