Question

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 a specific drug involved only 8 subjects. assume that we want to treat 8 healthy humans with this new drug and we have 12 suitable volunteers available. complete parts (a) through (c) below.
a. if the subjects are selected and treated in sequence, so that the trial is discontinued if anyone displays adverse effects, how many different sequential arrangements are possible if 8 people are selected from the 12 that are available? choose the correct answer below.
a. 40,320
b. 479,001,600
c. 19,958,400
d. 495
b. if 8 subjects are selected from the 12 that are available, and the 8 selected subjects are all treated at the same time, how many different treatment groups are possible?
there are 495 different treatment groups possible.
c. if 8 subjects are randomly selected and treated at the same time, what is the probability of selecting the 8 youngest subjects?
p(selecting the 8 youngest subjects) = (type an integer or a simplified fraction.)

Explanation:

Step1: Calculate permutations for part a

We use the permutation formula $P(n,r)=\frac{n!}{(n - r)!}$, where $n = 12$ and $r=8$. So $P(12,8)=\frac{12!}{(12 - 8)!}=\frac{12!}{4!}=\frac{12\times11\times10\times9\times8\times7\times6\times5\times4!}{4!}=19958400$.

Step2: Calculate combinations for part b

We use the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, with $n = 12$ and $r = 8$. Since $C(12,8)=C(12,4)$ (because $C(n,r)=C(n,n - r)$), then $C(12,4)=\frac{12!}{4!(12 - 4)!}=\frac{12\times11\times10\times9\times8!}{4!\times8!}=\frac{12\times11\times10\times9}{4\times3\times2\times1}=495$.

Step3: Calculate probability for part c

The probability of selecting a particular combination (the 8 - youngest) out of all possible combinations is the reciprocal of the number of combinations. Since there is 1 favorable outcome (selecting the 8 - youngest) and 495 total possible combinations from part b, the probability $P=\frac{1}{495}$.

Answer:

a. C. 19,958,400
b. 495
c. $\frac{1}{495}$