Question

In Exercises 5–36, express all probabilities as fractions.

Corporate Officers and Committees The Digital Pet Rock Company was recently successfully funded via Kickstarter and must now appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO). It must also appoint a strategic planning committee with four different members. There are 10 qualified candidates, and officers can also serve on the committee.

a. How many different ways can the four officers be appointed?

b. How many different ways can a committee of four be appointed?

c. What is the probability of randomly selecting the committee members and getting the four youngest of the qualified candidates?

Short answer

a. The number of different ways in which the four officers can be appointed is 5040.

b. The number of different ways in which a committee of four can be appointed is 210.

c. The probability of selecting the youngest four candidates as committee members is$\frac{1}{210}$.

Step-by-step solution

Given information

For a committee, four officers at different positions (president, COO, CEO, CFO) and four members are to be selected from a total of 10 candidates.

Compute the number of ways four officers can be selected

(a)

Permutation rule is applicable for selecting r units from n. In this case, the selection is done without replacement and the order of selection is insignificant.

Formula:

${}^n{P}_r=\frac{n!}{\left(n-r\right)!}$

The total number of qualified candidates is 10. Sincethe positions and the candidates are unique, the concept of permutation is used. The selected officers have an order/hierarchy corresponding to the positions.

Here, the total number of officers is 4.

The number of available candidates is 10.

The total number of ways in which four officers can be selected from 10 candidates while considering the order is

$$\begin{gathered} {}^{10}{P}_4=\frac{10!}{\left(10-4\right)!} \\ =\frac{10\times 9\times 8\times 7\times 6!}{6!} \\ =5040 \end{gathered}$$

Therefore, the number of ways in which four officers can be appointed is 5040.

Compute the number of ways four committee members can be selected

(b)

The number of ways in which r units can be selected from n units without replacement and in any order is given by the combination rule:

${}^n{C}_r=\frac{n!}{\left(n-r\right)!r!}$

The committee members can be appointed to the same position, so the order does not matter.

Here, the total number of candidates from which the committee members can be selected is 10.

The number of members to be chosen for the committee is 4.

The number of ways in which four candidates can be selected in any order is computed below:

$$\begin{gathered} {}^{10}{C}_4=\frac{10}{\left(10-4\right)!\times 4!} \\ =\frac{10\times 9\times 8\times 7\times 6!}{6!\times 4!} \\ =210 \end{gathered}$$

Therefore, the number of different ways in which a committee of four can be appointed is 210.

Compute the probability 

(c)

The total number of ways in which four members can be selected is 210.

There is one way of selecting the four youngest candidates.

The probability of selecting the four youngest candidates is given as follows:

$$\begin{gathered} P\left(\text{selecting}\text{4}\text{youngest}\text{candidates}\right)=\frac{\text{Number}\text{of}\text{favourable}\text{ways}}{\text{Total}\text{number}\text{of}\text{ways}} \\ =\frac{1}{210} \end{gathered}$$

Therefore,the probability of selecting the four youngest candidates as the committee members is$\frac{1}{210}$.