Chapter 2, Problem 90SE

A certain legislative committee consists of 10 senators. A subcommittee of 3 senators is to be randomly selected.

1. a. How many different such subcommittees are there?
2. b. If the senators are ranked 1, 2, ..., 10 in order of seniority, how many different subcommittees would include the most senior senator?
3. c. What is the probability that the selected subcommittee has at least 1 of the 5 most senior senators?
4. d. What is the probability that the subcommittee includes neither of the two most senior senators?

To determine

Find the number of different subcommittees.

Answer to Problem 90SE

The number of different subcommittees is 120.

Explanation of Solution

Given info:

A subcommittee consists of 3 senators who are randomly selected from the legislative committee of 10 senators.

Calculation:

Define the event:

$\begin{array}{l}N:\text{ Number of senators in the legislative committee}\\ n:\text{Number of senators in the subcommittee}\end{array}$

Combination:

The number of different arrangement of n elements from a set with N element is denoted as $\left({}_n^N\right)$ and is given by;

$\left({}_n^N\right)=\frac{N!}{\left(n!\right)\times \left(N-n\right)!}$

Substitute 10 for “N” and 3 for “n” in the above formula.

$\begin{array}{c}\left(\begin{array}{c}10\\ 3\end{array}\right)=\frac{10!}{3!\times (10-3)!}\\ =\frac{10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{\left(3\times 2\times 1\right)\times \left(7\times 6\times 5\times 4\times 3\times 2\times 1\right)}\\ =10\times 3\times 4\\ =120\end{array}$

Thus, the number of subcommittees is 120.

To determine

Find the number of subcommittee with senior senator.

Answer to Problem 90SE

The number of subcommittee with senior senator is 36.

Explanation of Solution

Given info:

The senator was ranked in the order of seniority. That is, 1, 2, 3, …, 10.

Calculation:

The senators were selected based on their seniority. So, one senior most senator will be selected and 2 senators have to be selected from remaining 9 senators.

The number of subcommittee with senior senator is obtained as shown below:

Substitute 9 for “N” and 2 for “n” in the above formula.

$\begin{array}{c}1\times \left(\begin{array}{c}9\\ 2\end{array}\right)=1\times \left(\frac{9!}{9!\times (9-2)!}\right)\\ =1\times \left(\frac{9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{\left(2\times 1\right)\times \left(7\times 6\times 5\times 4\times 3\times 2\times 1\right)}\right)\\ =9\times 4\\ =36\end{array}$

Thus, the number of subcommittee with senior senator is 36.

To determine

Find the probability of selecting the subcommitteewithat least 1 most senior senator.

Answer to Problem 90SE

The probability of selecting the subcommitteewithat least 1 most senior senatoris 0.9167.

Explanation of Solution

Calculation:

The total number of subcommittees is 120.

The number of most senior senators is 5.

Here, all 3 members are selected from most junior senators. The number of subcommittees includes none of the 5 most senior senators.

Substitute 5 for “N” and 3 for “n” in the above formula.

$\begin{array}{c}\left(\begin{array}{c}5\\ 3\end{array}\right)=\frac{5!}{3!\times (5-3)!}\\ =\frac{5\times 4\times 3\times 2\times 1}{\left(3\times 2\times 1\right)\times \left(2\times 1\right)}\\ =5\times 2\\ =10\end{array}$

Therefore, the number of subcommitteewithat least 1 most senior senatoris obtained as:

$\begin{array}{c}\text{at}\text{least}\text{one}\text{senior}\text{senator}=\left\{\begin{array}{l}\text{number of subcommitees}-\\ \text{number of senior senators }\\ \text{with none of senior senator}\end{array}\right\}\\ =120-10\\ =110\end{array}$

The probability of the subcommitteewithat least 1 most senior senatoris obtained as:

$\begin{array}{c}P\left(\text{at least one senior senator}\right)=\frac{\text{Number of senior senator}}{\text{Total number of possible subcommitee}}\\ =\frac{110}{120}\\ =0.9167\end{array}$

Thus the probability of the subcommittee with at least 1 most senior senator is 0.9167.

To determine

Find the probability that the subcommittee includes neither of the two senior senators.

Answer to Problem 90SE

The probability that the subcommittee includes neither of the two senior senators is 0.4667.

Explanation of Solution

Calculation:

The total number of subcommittees is 120.

The senators were selected based on their seniority. So, twomost senior senator will be selected and neither two of the senators should include in the subcommittee selection.

The number of subcommittee selected from the 8 senatorsis obtained as shown below:

Substitute 8 for “N” and 3 for “n” in the above formula.

$\begin{array}{c}\left(\begin{array}{c}8\\ 3\end{array}\right)=\left(\frac{8!}{3!\times (8-3)!}\right)\\ =\left(\frac{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{\left(3\times 2\times 1\right)\times \left(5\times 4\times 3\times 2\times 1\right)}\right)\\ =8\times 7\\ =56\end{array}$

The probability that the subcommittee includes neither of the two most senior senator is obtained as:

$\begin{array}{c}P\left(\text{neither of the two senior senator}\right)=\frac{\text{Number of subcommittees from 8 senators}}{\text{Total number of possible subcommitee}}\\ =\frac{56}{120}\\ =0.4667\end{array}$

Thus the probability that the subcommittee includes neither of the two most senior senator is 0.4667.