Precal Ch 9 Discrete Math

1. Count the number of ways that each procedure can be done. Line up 4 people (Mr. Han, Ms. Aguirre, Ms. Torres, and Mr. Birko) for a photograph.

There are 4 people to be lined up for the photograph. The first person can be any of the 4 individuals. After the first person is chosen, the second person can be any of the remaining 3 individuals. Similarly, the third person can be chosen from the remaining 2 individuals, and the last person will be the only remaining individual. Therefore, the total number of ways to line up the 4 people is 4 x 3 x 2 x 1 = 24.

Explanation

There are 4 people to be lined up for the photograph. The first person can be any of the 4 individuals. After the first person is chosen, the second person can be any of the remaining 3 individuals. Similarly, the third person can be chosen from the remaining 2 individuals, and the last person will be the only remaining individual. Therefore, the total number of ways to line up the 4 people is 4 x 3 x 2 x 1 = 24.

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2. A _______ graph is a graph with no edges.

In graph theory, a null graph is a graph that consists only of vertices (also called nodes) but has no edges connecting them. This means there are no relationships or connections between any of the vertices in the graph. Null graphs are a fundamental concept in graph theory and serve as a starting point for understanding more complex graph structures. They are also used in various applications, such as modeling systems with no interactions between their components.

Explanation

In graph theory, a null graph is a graph that consists only of vertices (also called nodes) but has no edges connecting them. This means there are no relationships or connections between any of the vertices in the graph. Null graphs are a fundamental concept in graph theory and serve as a starting point for understanding more complex graph structures. They are also used in various applications, such as modeling systems with no interactions between their components.

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3. Solve the equation algebraically. x = ?

The equation x = -9/5 can be solved algebraically by isolating the variable x on one side of the equation. In this case, x is already isolated, so the solution is x = -9/5. The answer -9/ 5 is the same as -9/5, just written in a different format.

Explanation

The equation x = -9/5 can be solved algebraically by isolating the variable x on one side of the equation. In this case, x is already isolated, so the solution is x = -9/5. The answer -9/ 5 is the same as -9/5, just written in a different format.

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4.

How many different answer keys are possible for a 10-question True-False Test?

There are 2 possible answers for each question (True or False). Since there are 10 questions in total, the number of different answer keys is calculated by multiplying the number of possible answers for each question together. Therefore, the total number of different answer keys is 2^10 = 1024.

Explanation

There are 2 possible answers for each question (True or False). Since there are 10 questions in total, the number of different answer keys is calculated by multiplying the number of possible answers for each question together. Therefore, the total number of different answer keys is 2^10 = 1024.

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5. Mr. Han gives 20 study questions, from which he will select 8 to be answered on the final exam. How many ways can he select the questions?

The answer is 125,970. This is because Mr. Han has 20 study questions and he needs to select 8 of them for the final exam. The number of ways he can select the questions is calculated using the combination formula, which is nCr = n! / (r!(n-r)!). In this case, n is 20 and r is 8. Plugging these values into the formula, we get 20! / (8!(20-8)!), which simplifies to 125,970.

Explanation

The answer is 125,970. This is because Mr. Han has 20 study questions and he needs to select 8 of them for the final exam. The number of ways he can select the questions is calculated using the combination formula, which is nCr = n! / (r!(n-r)!). In this case, n is 20 and r is 8. Plugging these values into the formula, we get 20! / (8!(20-8)!), which simplifies to 125,970.

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6.

How many different license plates consist of five symbols, either digits or letters?

The answer is 60,466,176. This is because for each symbol in the license plate, there are 36 possible options (26 letters + 10 digits). Therefore, for a license plate with 5 symbols, there are 36^5 = 60,466,176 different combinations possible.

Explanation

The answer is 60,466,176. This is because for each symbol in the license plate, there are 36 possible options (26 letters + 10 digits). Therefore, for a license plate with 5 symbols, there are 36^5 = 60,466,176 different combinations possible.

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7. The card game "bid Euchre" uses a pack of 24 cards, consisting of ace, king, queen, jack, 10, and 9 in each of the four suits (spades, hearts, diamonds, and clubs). In bid Euchre, a hand consists of 6 cards. Find the probability of each event. A hand is all spades.

The probability of getting a hand that consists of all spades can be calculated by dividing the number of favorable outcomes (hands with all spades) by the total number of possible outcomes (all possible hands).

In this case, there are 24 cards in the pack and a hand consists of 6 cards. Therefore, the total number of possible hands is given by the combination formula: C(24, 6) = 134,596.

Since there is only 1 favorable outcome (a hand with all spades), the probability of this event occurring is 1/134,596.

Explanation

The probability of getting a hand that consists of all spades can be calculated by dividing the number of favorable outcomes (hands with all spades) by the total number of possible outcomes (all possible hands).

In this case, there are 24 cards in the pack and a hand consists of 6 cards. Therefore, the total number of possible hands is given by the combination formula: C(24, 6) = 134,596.

Since there is only 1 favorable outcome (a hand with all spades), the probability of this event occurring is 1/134,596.

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8. Find the first 4 terms and the 20^th term of the sequences.

The given sequence is 0, 1, 2, 3, 19. It appears that the sequence follows a pattern where the first four terms are the numbers 0, 1, 2, and 3 in ascending order, and the fifth term is 19. Then, the pattern repeats with the next four terms being 0, 1, 2, and 3 again, followed by 19. Therefore, the 20th term of the sequence would also be 19.

Explanation

The given sequence is 0, 1, 2, 3, 19. It appears that the sequence follows a pattern where the first four terms are the numbers 0, 1, 2, and 3 in ascending order, and the fifth term is 19. Then, the pattern repeats with the next four terms being 0, 1, 2, and 3 again, followed by 19. Therefore, the 20th term of the sequence would also be 19.

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9.

The given answer is a combination of three different ways of writing the sequence -1, 2, 5, 8, 56. The first representation includes commas between each number. The second representation includes the word "and" before the last number. The third representation does not include any additional punctuation. Therefore, the answer includes all possible ways of writing the sequence.

Explanation

The given answer is a combination of three different ways of writing the sequence -1, 2, 5, 8, 56. The first representation includes commas between each number. The second representation includes the word "and" before the last number. The third representation does not include any additional punctuation. Therefore, the answer includes all possible ways of writing the sequence.

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11.

If it rains tomorrow, the probability is 0.8 that John will practice his piano lesson. If it does not rain tomorrow, there is only a 0.4 chance that John will practice. Suppose that the chance of rain tomorrow is 40%. What is the probability that John will practice his piano lesson? (Hint: Use a tree diagram)

The probability that John will practice his piano lesson can be calculated by multiplying the probability of rain (0.4) by the probability of practicing given rain (0.8) and adding it to the product of the probability of no rain (0.6) and the probability of practicing given no rain (0.4). This can be expressed as (0.4 * 0.8) + (0.6 * 0.4) = 0.32 + 0.24 = 0.56 or 56%.

Explanation

The probability that John will practice his piano lesson can be calculated by multiplying the probability of rain (0.4) by the probability of practicing given rain (0.8) and adding it to the product of the probability of no rain (0.6) and the probability of practicing given no rain (0.4). This can be expressed as (0.4 * 0.8) + (0.6 * 0.4) = 0.32 + 0.24 = 0.56 or 56%.

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12. A hand includes all four aces.

The answer 5/3542, 5/3,542 suggests that out of a total of 3,542 hands, only 5 of them include all four aces. This implies that the probability of getting a hand with all four aces is very low, as it only occurs in a small fraction of the total number of possible hands.

Explanation

The answer 5/3542, 5/3,542 suggests that out of a total of 3,542 hands, only 5 of them include all four aces. This implies that the probability of getting a hand with all four aces is very low, as it only occurs in a small fraction of the total number of possible hands.

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13. There are two precalculus sections at Roosevelt High School. There 12 girls and 8 boys in the 1^st period, while there are 10 girls and 15 boys in the 3^rd period. If a precalculus student chosen at random happens to be a girl, what is the probability she is from the 1^st period?

The probability that a precalculus student chosen at random is a girl from the 1st period can be found by dividing the number of girls in the 1st period by the total number of girls in both periods. In the 1st period, there are 12 girls, and in the 3rd period, there are 10 girls. Therefore, the total number of girls is 12 + 10 = 22. The probability is then 12/22, which simplifies to 6/11. This can be further simplified to 3/5, which is the given correct answer.

Explanation

The probability that a precalculus student chosen at random is a girl from the 1st period can be found by dividing the number of girls in the 1st period by the total number of girls in both periods. In the 1st period, there are 12 girls, and in the 3rd period, there are 10 girls. Therefore, the total number of girls is 12 + 10 = 22. The probability is then 12/22, which simplifies to 6/11. This can be further simplified to 3/5, which is the given correct answer.

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14. All six cards are from the same suit.

The answer provided is actually the same number written in two different formats. Both 1/33649 and 1/33,649 represent the probability of drawing all six cards from the same suit. The comma in 1/33,649 is used to separate thousands in some number systems, but it does not change the value of the fraction. Therefore, both answers are correct and equivalent.

Explanation

The answer provided is actually the same number written in two different formats. Both 1/33649 and 1/33,649 represent the probability of drawing all six cards from the same suit. The comma in 1/33,649 is used to separate thousands in some number systems, but it does not change the value of the fraction. Therefore, both answers are correct and equivalent.

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