Question

the offices of president, vice president, secretary, and treasurer for an environmental club will be filled from a pool of 13 candidates. eight of the candidates are members of the debate team.
(a) what is the probability that all of the offices are filled by members of the debate team?
(b) what is the probability that none of the offices are filled by members of the debate team?

(a) p(all offices filled by debate team members) =
(round to three decimal places as needed.)

Explanation:

Step1: Calculate total number of ways to fill offices

The number of ways to fill 4 offices from 13 candidates is given by permutation formula \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 13\) and \(r=4\).

$$P(13,4)=\frac{13!}{(13 - 4)!}=\frac{13!}{9!}=13\times12\times11\times10 = 17160$$

Step2: Calculate number of ways to fill offices with debate - team members

There are 8 debate - team members. The number of ways to fill 4 offices from 8 candidates is \(P(8,4)=\frac{8!}{(8 - 4)!}=\frac{8!}{4!}=8\times7\times6\times5=1680\)

Step3: Calculate the probability

The probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)

$$P=\frac{P(8,4)}{P(13,4)}=\frac{1680}{17160}\approx0.098$$

Answer:

\(0.098\)