Involve a deck of $52$ cards.

A poker hand consists of five cards.

$$\textbf{a.}\hspace{10pt}\text{Find the total number of possible five-card poker hands.}$$

$$\textbf{b.}\hspace{10pt}\text{Find the number of ways in which four aces can be selected.}$$

$$\textbf{c.}\hspace{10pt}\text{Find the number of ways in which one king can be selected.}$$

$$\textbf{d.}\hspace{10pt}\text{Use the Fundamental Counting Principle and your answers from parts (b) and (c) to find the number of ways getting four aces and one king.}$$

$$\textbf{e.}\hspace{10pt}\text{Find the probability of getting a poker hand of four aces and one king.}$$

Use the theoretical probability formula to solve each exercise. Express each probability as a fraction reduced to lowest terms.

You are dealt one card from a standard $52$-card deck. Find the probability of being dealt

a picture card.

the household income is not in the $\$ 15,000-\$ 24,999$ range.

Use the formula for $_nC_r$ to evaluate each expression.

$$_4C_4$$

Use the formula for $_{n} C_{r}$ to evaluate each expression. $_{25} C_{4}$

From the earlier exercise, find the probability of choosing a. a Democrat who is not a business major; b. a student who is neither a Democrat nor a business major.

Involve computing expected values in games of chance. A game is played using one die. If the die is rolled and shows 1, the player wins $5. If the die shows any number other than 1, the player wins nothing. If there is a charge of$1 to play the game, what is the game’s expected value? What does this value mean?

The table shows the distribution, by age, of a random sample of 3000 American moviegoers ages 12 through 74. Use this distribution.

$$\begin{matrix} \text{Ages} & \text{Nummber}\\ \text{43823} & \text{900}\\ \text{25-44} & \text{1080}\\ \text{45-64} & \text{840}\\ \text{65-74} & \text{180}\\ \end{matrix}$$

If one moviegoer is randomly selected from this population, find the probability, expressed as a simplified fraction, that the moviegoer’s age is at least 25.

Evaluate each factorial expression.

$$\dfrac{12!}{10!}$$

A single die is rolled twice. Find the probability of rolling

an odd number the first time and a number less than $3$ the second time.

Use the formula for $_nC_r$ to evaluate each expression.

$$_7C_7$$

A car model comes in nin colors, with or without air conditioning, with or without a sun roof, with or without automatic transmission, and with or without antilock brakes. In how many ways can the car be ordered with regard to these options?

You are taking a multiple-choice test that has eight questions. Each of the questions has three answer choices, with one correct answer per question. If you select one of these three choices for each question and leave nothing blank, in

Involve a deck of $52$ cards.

A poker hand consists of five cards.

$$\textbf{a.}\hspace{10pt}\text{Find the total number of possible five-card poker hands.}$$

$$\textbf{b.}\hspace{10pt}\text{A diamond flush is a five-card hand consisting of all diamonds. Find the number of possible diamond flushes.}$$

$$\textbf{c.}\hspace{10pt}\text{Find the probability of being health a diamond flush.}$$