Question

a corporation must appoint a president, chief executive officer (ceo), chief operating officer (coo), and chief financial officer (cfo). it must also appoint a planning committee with three different members. there are 12 qualified candidates, and officers can also serve on the committee. complete parts (a) through (c) below.
a. how many different ways can the four officers be appointed?
there are different ways to appoint the four officers.
b. how many different ways can a committee of three be appointed?
there are different ways to appoint a committee of three.
c. what is the probability of randomly selecting the committee members and getting the three youngest of the qualified candidates?
p(getting the three youngest of the qualified candidates)= (type an integer or a simplified fraction.)

Explanation:

Step1: Calculate number of ways to appoint 4 officers

The number of permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\). Here \(n = 12\) and \(r=4\). So \(P(12,4)=\frac{12!}{(12 - 4)!}=\frac{12!}{8!}=12\times11\times10\times9 = 11880\).

Step2: Calculate number of ways to appoint 3 - member committee

The number of combinations of \(n\) objects taken \(r\) at a time is \(C(n,r)=\frac{n!}{r!(n - r)!}\). Here \(n = 12\) and \(r = 3\). So \(C(12,3)=\frac{12!}{3!(12 - 3)!}=\frac{12\times11\times10\times9!}{3!\times9!}=\frac{12\times11\times10}{3\times2\times1}=220\).

Step3: Calculate probability of getting 3 youngest candidates

There is only 1 way to select the 3 youngest candidates out of the total 220 ways of selecting a 3 - member committee. So the probability \(P=\frac{1}{220}\).

Answer:

a. 11880
b. 220
c. \(\frac{1}{220}\)