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20 Jan 2024, 07:15
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E
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Difficulty:
95%
(hard)
Question Stats:
29% (02:48) correct
71%
(02:54)
wrong
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A book publisher has 4 horror stories, 3 romantic stories, and 2 comedy stories for his upcoming book. If the book must include at least one romantic story and at least one comedy story, and the order of the stories in the book does not matter, how many different books can he publish?
A. 6
B. 21
C. 96
D. 128
E. 336
A. 6
B. 21
C. 96
D. 128
E. 336
20 Jan 2024, 08:21
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Bunuel
For each story that the author has, he has two choices - either he can select the story for publishing or not select the story for publishing.
- Number of choices available for four horror stories= \(2 * 2 * 2 * 2 = 2^4 = 16\)
- Number of choices available for three romantic stories = \(2 * 2 * 2 = 2^3 = 8\). However, among the 8 choices, the author may choose not to select any romantic story. We have to remove this choice as the book must include at least one romantic story.
Number of choices available to the author so that he chooses at least one romantic story = \( 8 - 1 = 7\) - Number of choices available for two comedy stories = \(2 * 2 = 2^2 = 4\). However, among the 4 choices, the author may choose not to select any comedy story. We have to remove this choice as the book must include at least one comedy story.
Number of choices available to the author so that he chooses at least one comedy story = \( 4 - 1 = 3\)
Number of books that the author can publish = \(16 * 7 * 3 = 16 * 21 = 336\)
Option E
22 Jan 2024, 02:24
Kudos
Bookmarks
A book publisher has 4 horror stories, 3 romantic stories, and 2 comedy stories for his upcoming book. If the book must include at least one romantic story and at least one comedy story, and the order of the stories in the book does not matter, how many different books can he publish?
A. 6
B. 21
C. 96
D. 128
E. 336
Each of the 4 horror stories has two options - either to be included in the book or not. Hence, the total number of ways to represent the 4 horror stories in the book is \(2^4\). Note that this count includes the scenario where none of the horror stories is included.
Similarly, each of the 3 romantic stories has two options - either to be included in the book or not. Hence, the total number of ways to represent the 3 romantic stories in the book is \(2^3\). However, since this count includes the scenario where none of the romantic stories is included, and we need at least one romantic story in the book, we should subtract that one case from \(2^3\). Thus, the total ways to represent the 3 romantic stories in the book, ensuring at least one is included, is \(2^3 - 1\).
Next, each of the 2 comedy stories has two options - either to be included in the book or not. Hence, the total number of ways to represent the 2 comedy stories in the book is \(2^2\). However, since this count also includes the scenario where none of the comedy stories is included, and we need at least one comedy story in the book, we should subtract that one case from \(2^2\). Thus, the total ways to represent the 2 comedy stories in the book, ensuring at least one is included, is \(2^2 - 1\).
Therefore, the total number of ways to represent these stories in the book is \(2^4(2^3 - 1)(2^2 - 1) = 336\).
Answer: E
A. 6
B. 21
C. 96
D. 128
E. 336
Each of the 4 horror stories has two options - either to be included in the book or not. Hence, the total number of ways to represent the 4 horror stories in the book is \(2^4\). Note that this count includes the scenario where none of the horror stories is included.
Similarly, each of the 3 romantic stories has two options - either to be included in the book or not. Hence, the total number of ways to represent the 3 romantic stories in the book is \(2^3\). However, since this count includes the scenario where none of the romantic stories is included, and we need at least one romantic story in the book, we should subtract that one case from \(2^3\). Thus, the total ways to represent the 3 romantic stories in the book, ensuring at least one is included, is \(2^3 - 1\).
Next, each of the 2 comedy stories has two options - either to be included in the book or not. Hence, the total number of ways to represent the 2 comedy stories in the book is \(2^2\). However, since this count also includes the scenario where none of the comedy stories is included, and we need at least one comedy story in the book, we should subtract that one case from \(2^2\). Thus, the total ways to represent the 2 comedy stories in the book, ensuring at least one is included, is \(2^2 - 1\).
Therefore, the total number of ways to represent these stories in the book is \(2^4(2^3 - 1)(2^2 - 1) = 336\).
Answer: E
