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:
(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.
(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

Explanation:

Step1: Calculate total number of ways to form committee

Use combination formula \(C(n,k)=\frac{n!}{k!(n - k)!}\). Total people \(n=21 + 6=27\), committee size \(k = 4\).
\(C(27,4)=\frac{27!}{4!(27-4)!}=\frac{27\times26\times25\times24}{4\times3\times2\times1}=17550\)

Step2: Calculate number of ways to choose 2 students and 2 faculty

Number of ways to choose 2 students from 21: \(C(21,2)=\frac{21!}{2!(21 - 2)!}=\frac{21\times20}{2\times1} = 210\)
Number of ways to choose 2 faculty from 6: \(C(6,2)=\frac{6!}{2!(6 - 2)!}=\frac{6\times5}{2\times1}=15\)
By multiplication principle, number of ways to form 2 - student and 2 - faculty committee is \(C(21,2)\times C(6,2)=210\times15 = 3150\)

Step3: Calculate probability

Probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)
\(P=\frac{3150}{17550}\approx0.17949\)

Answer:

\(0.17949\)