In Exercises 5–36, express all probabilities as fractions.

Phase I of a Clinical Trial A clinical test on humans of a new drug is normally done in three phases. Phase I is conducted with a relatively small number of healthy volunteers. For example, a phase I test of bexarotene involved only 14 subjects. Assume that we want to treat 14 healthy humans with this new drug and we have 16 suitable volunteers available.

a. If the subjects are selected and treated one at a time in sequence, how many different sequential arrangements are possible if 14 people are selected from the 16 that are available?

b. If 14 subjects are selected from the 16 that are available, and the 14 selected subjects are all treated at the same time, how many different treatment groups are possible?

c. If 14 subjects are randomly selected and treated at the same time, what is the probability of selecting the 14 youngest subjects?

Fourteen people are to be selected from a group of 16 people who volunteered for a clinical trial.

The number of different ways in which r items can be selected from n items without replacement,when the sequence of the selected units matters, is given by the permutation rule:

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

The number of different ways in which r items can be selected from n items without replacement,when the sequence of the selected units does not matter, is given by the combination rule:

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

a.

When the subjects are treated one by one, their order of selection matters. Thus, the permutation rule is used.

Here, the total number of volunteers is 16.

The number of people to be selected is 14.

The number of different sequential arrangements possible when 14 are selected from a group of 16 people is:

$$\begin{gathered} {}^{16}{P}_{14}=\frac{16!}{\left(16-14\right)!} \\ =\frac{16!}{2!} \\ =10,461,394,944,000. \end{gathered}$$

Therefore, the number of different sequential arrangements possible when 14 are selected from a group of 16 people is 10,461,394,944,000.

b.

When the subjects are handled at a time, the order of treatment does not matter. Thus, the combination rule is used.

Here, the total number of people is 16.

The number of people to be selected is 14.

The number of different treatment groups possible when 14 are selected from a group of 16 people (in any order) is

$$\begin{gathered} {}^{16}{C}_{14}=\frac{16!}{\left(16-14\right)!14!} \\ =120 \end{gathered}$$

Therefore, the number of different treatment groups possible when 14 people are selected from a group of 16 people is 120.

c.

The number of ways in which the 14 youngest people can be selected is 1.

The total number of ways of selecting 14 people is 120.

The probability of selecting the 14 youngest people is

$$\begin{gathered} P\left(14\text{youngest}\text{people}\right)=\frac{1}{120} \\ =0.0833 \end{gathered}$$

Therefore, the probability of selecting the 14 youngest people is 0.0833.