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 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 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 officer - appointment permutations

The number of ways to appoint 4 officers out of 11 candidates (where order matters) is given by the permutation formula $P(n,r)=\frac{n!}{(n - r)!}$, with $n = 11$ and $r=4$.
$P(11,4)=\frac{11!}{(11 - 4)!}=\frac{11!}{7!}=11\times10\times9\times8 = 7920$

Step2: Calculate committee - appointment combinations

The number of ways to appoint a committee of 3 out of 11 candidates (where order does not matter) is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, with $n = 11$ and $r = 3$.
$C(11,3)=\frac{11!}{3!(11 - 3)!}=\frac{11\times10\times9\times8!}{3\times2\times1\times8!}=165$

Step3: Calculate the probability

The probability of randomly selecting the three youngest candidates for the committee is the number of favorable outcomes (which is 1, since there is only one group of the three youngest candidates) divided by the total number of possible outcomes (165). So $P=\frac{1}{165}$

Answer:

a. 7920
b. 165
c. $\frac{1}{165}$