Chapter 11.1, Problem 54PPSE

Solution

a.

To calculate: The number of possible combinations of five people who squeeze toothpaste from the middle of the tube.

The resultant answer is 8,568 ways.

Given information:

The given bar graph shows the results of 40 responses to a survey:

  

Formula used: If $C{}_r$ represents the number of combinations of n items taken r at a time, then $C{}_r=\frac{n!}{\left(n-r\right)!r!}$ .

Calculation:

From the graph, it can be seen that 45 percent of 40 squeeze toothpaste from the middle:

  $\frac{45}{100}\times 40=18\text{people}$

The number of ways in which 5 people can be selected from a group of 18 people,

  $C{}_5$

Rewrite the expression using the formula $C{}_r=\frac{n!}{\left(n-r\right)!r!}$ ,

  $\begin{array}{c}C{}_5=\frac{18!}{\left(18-5\right)!5!}\\ =\frac{18!}{13!5!}\end{array}$

Now expand the factorial,

  $\frac{18\times 17\times 16\times 15\times 14\times 13!}{13!\left(5\times 4\times 3\times 2\times 1\right)}=8568$

Therefore, there are 8568 number of ways in which 5 people can be selected from a group of 18 people.

b.

To calculate: The number of possible combinations of five people if five people are chosen at random from all the people who responded to the survey.

The resultant answer is $658,008$ .

Given information:

Total people participated in the survey is 40.

Formula used: If $C{}_r$ represents the number of combinations of n items taken r at a time, then $C{}_r=\frac{n!}{\left(n-r\right)!r!}$ .

Calculation:

The number of ways in which 5 people can be selected from a group of 40 people,

  $C{}_5$

Rewrite the expression using the formula $C{}_r=\frac{n!}{\left(n-r\right)!r!}$ ,

  $\begin{array}{c}C{}_5=\frac{40!}{\left(40-5\right)!5!}\\ =\frac{40!}{35!5!}\end{array}$

Now expand the factorial,

  $\frac{40\times 39\times 38\times 37\times 36\times 35!}{35!\left(5\times 4\times 3\times 2\times 1\right)}=658,008$

Therefore, there are $658,008$ number of ways in which 5 people can be selected from a group of 40 people.