Chapter 5.4, Problem 34E

Committee: The Student Council at a certain school has 10 members. Four members will form an executive committee consisting of a president, a vice president, a secretary, and a treasurer.

In how many ways can these four positions be filled?

1. In how many ways can four people be chosen for due executive committee if it does not matter who gets which position?
2. Four of the people on Student Council are Zachary, Yolanda, Xavier, and Walter. What is the probability that Zachary is president, Yolanda is vice president, Xavier is secretary, and Walter is treasurer?
3. What is the probability that Zachary, Yolanda, Xavier and Walter are due four committee members?

To determine

To find:The number of ways the given four positions can be filled.

Answer to Problem 34E

The possible number is $\underset{\_}{5,040}.$

Explanation of Solution

Given information:

The Student Councilat a certain school has ten members. Four members will form an executive committee consisting of a president, a vice president, a secretary, and a treasurer.

Concept Used:

We have the fundamental counting principle which states that if there are three events that can occur in m, n and p ways respectively then the total number of ways in which all three events can occur is $m.n.p$ .

We know that the permutations of objects are the arrangements of the objects. And in the arrangement of objects, the order of objects does matter. For example, if we have three objects say, A, B, and C, there are total of six arrangements of the objects that are possible and these areABC, ACB, BAC, BCA, CAB, and CBA.

We know that combination is a selection of r objects out of n different objects and the order of the selection of objects does not matter. For example, if there are three objects say, P, Q, and R and we have to select two objects out of three. Then while selection, selecting P and then Q is the same as selecting Q and then P. So, in other words selecting PQ is the same as selecting QP.

Calculation:

So, according to the question, the president can be selected in 10 ways, vice-president can be selected in 9 ways, the secretary can be selected in 8 ways and treasurer can be selected in 7 ways. So,the possible number of different committees is;

  $\mathrm{10.9.8.7}=5040$

Hence, the possible number of different committees is $\underset{\_}{5,040}.$

To determine

To find:The number ofways four people can be chosen for the executive committee if it does not matter who gets which position.

Answer to Problem 34E

The possible number is $\underset{\_}{210}.$

Explanation of Solution

Given information:

The Student Councilat a certain school has ten members. Four members will form an executive committee consisting of a president, a vice president, a secretary, and a treasurer.

Concept Used:

We have the fundamental counting principle which states that if there are three events that can occur in m, n and p ways respectively then the total number of ways in which all three events can occur is $m.n.p$ .

We know that the permutations of objects are the arrangements of the objects. And in the arrangement of objects, the order of objects does matter. For example, if we have three objects say, A, B, and C, there are a total of six arrangements of the objects that are possible and these are; ABC, ACB, BAC, BCA, CAB, and CBA.

We know that combination is a selection of r objects out of n different objects and the order of the selection of objects does not matter. For example, if there are three objects say, P, Q, and R and we have to select two objects out of three. Then while selection, selecting P and then Q is the same as selecting Q and then P. So, in other words selecting PQ is the same as selecting QP.

Calculation:

So according to the question, the possible number of ways of selecting 4 members out of 10 members is calculated using a combination;

  $\begin{array}{l}C{}_4=\frac{10!}{4!\left(10-4\right)!}\\ =\frac{10!}{4!\left(6!\right)}=\frac{\mathrm{10.9.8.7.6}!}{\mathrm{4.3.2.1}\left(6!\right)}=210\\ \Rightarrow C{}_4=210\end{array}$

The possible number is $\underset{\_}{210}.$

To determine

To find:If four of the people on the Student Council are Zachary, Yolanda, Xavier, and Walter then the probability that Zachary is president, Yolanda is vice-president, Xavier is secretary and Walter is treasurer.

Answer to Problem 34E

The required probability is $\underset{\_}{0.000198}.$

Explanation of Solution

Given information:

The Student Councilat a certain school has ten members. Four members will form an executive committee consisting of a president, a vice president, a secretary, and a treasurer.

Concept Used:

We have the fundamental counting principle which states that if there are three events that can occur in m, n and p ways respectively then the total number of ways in which all three events can occur is $m.n.p$ .

We know that the permutations of objects are the arrangements of the objects. And in the arrangement of objects, the order of objects does matter. For example, if we have three objects say, A, B, and C, there are a total of six arrangements of the objects are possible and these are; ABC, ACB, BAC, BCA, CAB, and CBA.

We know that combination is a selection of r objects out of n different objects and order o the selection of objects does not matter. For example, if there are three objects say, P, Q, and R and we have to select two objects out of three. Then while selection, selecting P, and then Q is the same as selecting Q and then P. So, in other words selecting PQ is the same as selecting QP.

Calculation:

According to the question, there is only one way in which these 4 members can be selected in the given order. Thus, the probability of the required committee is;

  $\begin{array}{l}P=\frac{1}{5040}\\ =0.000198\\ \Rightarrow P=0.000198\end{array}$

Conclusion:

The required probability is $\underset{\_}{0.000198}.$

To determine

To find:The probability that Zachary, Yolanda, Xavier and Walter are the four committee members.

Answer to Problem 34E

The required probability is $\underset{\_}{0.0048}.$

Explanation of Solution

Given information:

The Student Councilat a certain school has ten members. Four members will form an executive committee consisting of a president, a vice president, a secretary, and a treasurer.

Concept Used:

We have the fundamental counting principle which states that if there are three events that can occur in m, n and p ways respectively then the total number of ways in which all three events can occur is $m.n.p$ .

We know that the permutations of objects are the arrangements of the objects. And in the arrangement of objects, the order of objects does matter. For example, if we have three objects say, A, B, and C, there are a total of six arrangements of the objects that are possible and these are; ABC, ACB, BAC, BCA, CAB, and CBA.

We know that combination is a selection of r objects out of n different objects and order o the selection of objects does not matter. For example, if there are three objects say, P, Q, and R and we have to select two objects out of three. Then while selection, selecting P and then Q is the same as selecting Q and then P. So, in other words selecting PQ is the same as selecting QP.

Calculation:

According to the question, there is only one way in which these 4 members can be selected. Thus, the probability of the required committee is;

  $\begin{array}{l}P=\frac{1}{210}\\ =0.0048\\ \Rightarrow P=0.0048\end{array}$

Hence, the required probability is $\underset{\_}{0.0048}.$