Question

a student - faculty government committee of 4 people is to be formed from 21 student volunteers and 6 faculty volunteers. find the probability that the committee will consist of the following, assuming the selection is made at random:
part 1 of 4
(a) all faculty members. round your answer to five decimal places.
the probability that the committee will consist of all faculty members is 0.00085.
part 2 of 4
(b) two students and two faculty members. round your answer to five decimal places.
the probability that the committee will consist of two students and two faculty members is 0.17949.
part: 2 / 4
part 3 of 4
(c) all students. round your answer to five decimal places.
the probability that the committee will consist of all students is

Explanation:

Step1: Calculate the total number of ways to form the committee

The total number of people is \(21 + 6=27\). We need to choose \(4\) people for the committee. Using the combination formula \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 27\) and \(r=4\).

$$C(27,4)=\frac{27!}{4!(27 - 4)!}=\frac{27\times26\times25\times24}{4\times3\times2\times1}=17550$$

Step2: Calculate the number of ways to form a committee of all students

We have \(21\) students and we need to choose \(4\) of them. Using the combination formula with \(n = 21\) and \(r = 4\)

$$C(21,4)=\frac{21!}{4!(21 - 4)!}=\frac{21\times20\times19\times18}{4\times3\times2\times1}=5985$$

Step3: Calculate the probability

The probability \(P\) that the committee consists of all students is given by the ratio of the number of favorable outcomes (choosing 4 students) to the number of total outcomes (choosing 4 people from 27)

$$P=\frac{C(21,4)}{C(27,4)}=\frac{5985}{17550}= 0.341025641\approx0.34103$$

Answer:

\(0.34103\)