Chapter 7

A state Motor Vehicle Department requires learners to pass a written test on the motor vehicle laws of the state. The exam consists of ten true-or-false questions, of which eight must be answered correctly to qualify for a permit. In how many different ways can a learner who answers all the questions on the exam qualify for a permit?

A list of poker hands ranked in order from the highest to the lowest is shown in the following table, along with a description and example of each hand. Use the table to answer. If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, how many different hands consist of the following: A straight flush? (Note that an ace may be played as either a high or a low card in a straight sequence- that is, A, 2,3 , 4, 5 or \(10, \mathrm{~J}, \mathrm{Q}, \mathrm{K}, \mathrm{A}\). Hence, there are ten possible sequences for a straight in one suit.)

A list of poker hands ranked in order from the highest to the lowest is shown in the following table, along with a description and example of each hand. Use the table to answer. If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, how many different hands consist of the following: A flush (but not a straight flush)?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. $$ \left(A \cup A^{c}\right)^{c}=\varnothing $$

A list of poker hands ranked in order from the highest to the lowest is shown in the following table, along with a description and example of each hand. Use the table to answer. If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, how many different hands consist of the following: Four of a kind?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A \subseteq B\), then \(A \cap B=A\).

A list of poker hands ranked in order from the highest to the lowest is shown in the following table, along with a description and example of each hand. Use the table to answer. If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, how many different hands consist of the following: A full house?

A list of poker hands ranked in order from the highest to the lowest is shown in the following table, along with a description and example of each hand. Use the table to answer. If a 5-card poker hand is dealt from a well-shuffled deck of 52 cards, how many different hands consist of the following: Two pair?

In the World Series, one National League team and one American League team compete for the title, which is awarded to the first team to win four games. In how many different ways can the series be completed?

A quorum (minimum) of 6 voting members is required at all meetings of the Curtis Townhomes Owners Association. If there is a total of 12 voting memhers in the group, find the numher of ways this quorum can be formed.

Suppose \(n\) distinct objects are arranged in a circle. Show that the number of (different) circular arrangements of the \(n\) objects is \((n-1) !\) Hint: Consider the arrangement of the five letters \(A, B, C, D\), and \(E\) in the accompanying figure. The permutations \(A B C D E, B C D E A\), \(C D E A B, D E A B C\), and \(E A B C D\) are not distinguishable. Generalize this observation to the case of \(n\) objects.

At the end of Section \(6.3\), we mentioned that solving a linear programming problem in three variables and five constraints by the methods of corners requires that we solve \(563 \times 3\) systems of linear equations. Verify this assertion.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The number of permutations of \(n\) distinct objects taken all together is \(n !\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The number of combinations of \(n\) objects taken \(n-r\) at a time is the same as the number taken \(r\) at a time.