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 five different members. there are 11 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 five be appointed?
there are different ways to appoint a committee of five.
c. what is the probability of randomly selecting the committee members and getting the five youngest of the qualified candidates?
p(getting the five youngest of the qualified candidates)= (type an integer or a simplified fraction.)

Explanation:

Step1: Calculate officer appointments

The number of permutations of \(n\) objects taken \(r\) at a time is \(P(n,r)=\frac{n!}{(n - r)!}\). Here \(n = 11\) and \(r=4\). So \(P(11,4)=\frac{11!}{(11 - 4)!}=\frac{11!}{7!}=11\times10\times9\times8 = 7920\).

Step2: Calculate committee appointments

The number of combinations of \(n\) objects taken \(r\) at a time is \(C(n,r)=\frac{n!}{r!(n - r)!}\). Here \(n = 11\) and \(r = 5\). So \(C(11,5)=\frac{11!}{5!(11 - 5)!}=\frac{11\times10\times9\times8\times7}{5\times4\times3\times2\times1}=462\).

Step3: Calculate probability

There is only 1 way to get the five - youngest candidates out of \(C(11,5)\) possible combinations. So the probability \(P=\frac{1}{462}\).

Answer:

a. 7920
b. 462
c. \(\frac{1}{462}\)