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.

$\textbf{a.}\hspace{10pt}$ In how many ways can these four positions be filled?

$\textbf{b.}\hspace{10pt}$ In how many ways can four people be chosen for the executive committee if it does not matter who gets which position?

$\textbf{c.}\hspace{10pt}$ 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?

$\textbf{d.}\hspace{10pt}$ What is the probability that Zachary, Yolanda, Xavier, and Walter are the four committee members?

Fifteen items or less: The number of customers in line at a supermarket express checkout counter is a random variable with the following probability distribution.

$$\begin{array}{c|cccccc} x & 0 & 1 & 2 & 3 & 4 & 5\\ \hline P(x) & 0.10 & 0.25 & 0.30 & 0.20 & 0.10 & 0.05\\ \end{array}$$

$\textbf{a.}\hspace{10pt}$ Find $P(2)$.

$\textbf{b.}\hspace{10pt}$ Find $P(\text{No more than}\ 1)$.

$\textbf{c.}\hspace{10pt}$ Find the probability that no one is in line.

$\textbf{d.}\hspace{10pt}$ Find the probability that at least three people are in line.

$\textbf{e.}\hspace{10pt}$ Compute the mean $\mu_x$.

$\textbf{f.}\hspace{10pt}$ Compute the standard deviation $\sigma_x$.

$\textbf{g.}\hspace{10pt}$ If each customer takes $3$ minutes to check out, what is the probability that it will take more than $6$ minutes for all the customers currently in line to check out?

Write the given number below as a power of 10 .

10,000,000,000,000

An executive committee consisted of 10 members: 4 women and 6 men. Three members were selected at random to be sent to a meeting in Hawaii. A blindfolded woman drew 3 of the 10 names from a hat to determine who would go. All 3 names drawn were women’s. What was the probability of such luck?

A student council consists of three freshmen, four sophomores, four juniors, and five seniors. How many committees of eight members of the council contain at least one member from each class?

The number of customers in line at a supermarket express checkout counter is a random variable whose probability mass function is given in the following table.

$$\begin{matrix} \text{x} & \text{0} & \text{1} & \text{2} & \text{3} & \text{4} & \text{5}\\ \text{p(x)} & \text{0.10} & \text{0.25} & \text{0.30} & \text{0.20} & \text{0.10} & \text{0.05}\\ \end{matrix}$$

For each customer, the number of items to be purchased is a random variable with probability mass function

$$\begin{matrix} \text{y} & \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6}\\ \text{p(y)} & \text{0.05} & \text{0.15} & \text{0.25} & \text{0.30} & \text{0.15} & \text{0.10}\\ \end{matrix}$$

Let X denote the number of customers in line, and let Y denote the total number of items purchased by all the customers in line. Assume the number of items purchased by one customer is independent of the number of items purchased by any other customer. a. Find P(X = 2 and Y = 2). b. Find P(X = 2 and Y = 6). c. Find P(Y = 2).

A club has 40 members and 10 pledges. The club selects a 5-member executive committee composed of a chair and vice-chair, who must be members, and an advisory committee of 3 pledges. In how many ways can the executive committee be formed?

A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a chairperson, secretary, and treasurer. How many selections are there in which Dolph is an officer and Francisco is not an officer?

In horse racing, one can make a trifecta bet by specifying which horse will come in first, which will come in second, and which will come in third, in the correct order. One can make a box trifecta bet by specifying which three horses will come in first, second, and third, without specifying the order. a. In an eight-horse field, how many different ways can one make a trifecta bet? b. In an eight-horse field, how many different ways can one make a box trifecta bet?

The table above lists the highest prices five consumers are willing to pay for a concert ticket. If the price of one of the tickets is$36

A) Violet and Walter will each buy two tickets.

B) Walter will receive $4 of consumer surplus from buying one ticket.

C) Violet and Walter receive a total of$52 of consumer surplus from buying one ticket each. No one else will buy a ticket.

D) Xavier, Yolanda, and Zachary will receive a total of $68 of consumer surplus since they will buy no tickets.

The table above lists the highest prices five consumers are willing to pay for a concert ticket. If the price of one of the tickets is$20

A) everyone will buy a ticket except for Zachary.

B) only Violet and Walter will buy tickets.

C) Xavier's consumer surplus is $50.

D) the total consumer surplus from the purchase of tickets will be$122.

The table above lists the highest prices five consumers are willing to pay for a concert ticket. If the price of one ticket rises from$20 to $38

A) only three tickets will be sold.

B) consumer surplus decreases from$62 to $12.

C) consumer surplus increases from$88 to $142.

D) no one will buy a ticket.

The qualified applicant pool for six management trainee positions consists of seven women and five men. If the applicants are equally qualified and the trainee positions are selected by drawing the names at random so that all groups of six are equally likely, what is the probability that the trainee class will consist entirely of women?

A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a chairperson, secretary, and treasurer. How many selections are there in which both Ben and Francisco are officers?