Question

the offices of president, vice president, secretary, and treasurer for an environmental club will be filled from a pool of 19 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(\text{all offices filled by debate team members}) = 0.018 )
(round to three decimal places as needed.)
(b) ( p(\text{no offices filled by debate team members})=square )
(round to three decimal places as needed.)

Explanation:

Step1: Calculate the total number of ways to fill the offices

The number of ways to choose \(4\) people out of \(19\) for the \(4\) different offices (permutation) is \(P(n,r)=\frac{n!}{(n - r)!}\), where \(n = 19\) and \(r=4\). So \(P(19,4)=\frac{19!}{(19 - 4)!}=\frac{19!}{15!}=19\times18\times17\times16=93024\)

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

There are \(19 - 8=11\) non - debate - team members. The number of ways to choose \(4\) people out of \(11\) for the \(4\) different offices (permutation) is \(P(11,4)=\frac{11!}{(11 - 4)!}=\frac{11!}{7!}=11\times10\times9\times8 = 7920\)

Step3: Calculate the probability

The probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). So \(P=\frac{P(11,4)}{P(19,4)}=\frac{7920}{93024}\approx0.085\)

Answer:

\(0.085\)