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Sagot :
Let's address each statement in detail and check their validity based on the provided numerical results.
### Statement 1:
"There are [tex]\({ }_{20} C _3\)[/tex] possible ways to choose three books to read."
To find the number of ways Mariah can choose three books from a total of 5 mysteries, 7 biographies, and 8 science fiction novels, first calculate the total number of books:
[tex]\[ 5 + 7 + 8 = 20 \][/tex]
The number of ways to choose 3 books out of 20 is given by the combination [tex]\({ }_{20} C_3\)[/tex].
From our results:
[tex]\[ \text{Number of ways} = 1140.0 \][/tex]
Hence, this statement is true.
### Statement 2:
"There are [tex]\({ }_5 C _3\)[/tex] possible ways to choose three mysteries to read."
The number of ways to choose 3 books out of 5 is given by the combination [tex]\({ }_5 C_3\)[/tex].
From our results:
[tex]\[ \text{Number of ways} = 10.0 \][/tex]
Hence, this statement is true.
### Statement 3:
"There are [tex]\({ }_{15} C_3\)[/tex] possible ways to choose three books that are not all mysteries."
To find the number of ways to choose 3 books that are not all mysteries, we need to exclude the mysteries from the total count:
[tex]\[ 7 \, (\text{biographies}) + 8 \, (\text{science fiction}) = 15 \][/tex]
The number of ways to choose 3 books out of 15 is given by the combination [tex]\({ }_{15} C_3\)[/tex].
From our results:
[tex]\[ \text{Number of ways} = 455.0 \][/tex]
Hence, this statement is true.
### Statement 4:
"The probability that Mariah will choose 3 mysteries can be expressed as [tex]\(\frac{1}{{ }_5 C_3}\)[/tex]."
First, calculate the number of ways to choose 3 mysteries out of 5:
[tex]\[ \text{Number of ways to choose 3 mysteries} = { }_5 C_3 = 10.0 \][/tex]
The probability that all 3 chosen books are mysteries is inversely proportional to the number of ways to choose 3 mysteries, which can be expressed as:
[tex]\[ \text{Probability} = \frac{1}{10.0} = 0.1 \][/tex]
Hence, this statement is true.
### Statement 5:
"The probability that Mariah will not choose all mysteries can be expressed as [tex]\(1 - \frac{{ }_{2} C_3}{{ }_{20} C_3}\)[/tex]."
Here, [tex]\({ }_{2} C_3\)[/tex] denotes the number of ways to choose 3 books, all of which are not mysteries. Since [tex]\({ }_2 C_3 = 0\)[/tex] (because you can't choose 3 out of 2), the expression becomes:
[tex]\[ \frac{0}{20} = 0 \][/tex]
Thus, the probability that all 3 chosen books are not mysteries can be computed as:
[tex]\[ 1 - 0 = 1.0 \][/tex]
Hence, this statement is true.
### Conclusion:
All the given statements are true based on the provided numerical results.
### Statement 1:
"There are [tex]\({ }_{20} C _3\)[/tex] possible ways to choose three books to read."
To find the number of ways Mariah can choose three books from a total of 5 mysteries, 7 biographies, and 8 science fiction novels, first calculate the total number of books:
[tex]\[ 5 + 7 + 8 = 20 \][/tex]
The number of ways to choose 3 books out of 20 is given by the combination [tex]\({ }_{20} C_3\)[/tex].
From our results:
[tex]\[ \text{Number of ways} = 1140.0 \][/tex]
Hence, this statement is true.
### Statement 2:
"There are [tex]\({ }_5 C _3\)[/tex] possible ways to choose three mysteries to read."
The number of ways to choose 3 books out of 5 is given by the combination [tex]\({ }_5 C_3\)[/tex].
From our results:
[tex]\[ \text{Number of ways} = 10.0 \][/tex]
Hence, this statement is true.
### Statement 3:
"There are [tex]\({ }_{15} C_3\)[/tex] possible ways to choose three books that are not all mysteries."
To find the number of ways to choose 3 books that are not all mysteries, we need to exclude the mysteries from the total count:
[tex]\[ 7 \, (\text{biographies}) + 8 \, (\text{science fiction}) = 15 \][/tex]
The number of ways to choose 3 books out of 15 is given by the combination [tex]\({ }_{15} C_3\)[/tex].
From our results:
[tex]\[ \text{Number of ways} = 455.0 \][/tex]
Hence, this statement is true.
### Statement 4:
"The probability that Mariah will choose 3 mysteries can be expressed as [tex]\(\frac{1}{{ }_5 C_3}\)[/tex]."
First, calculate the number of ways to choose 3 mysteries out of 5:
[tex]\[ \text{Number of ways to choose 3 mysteries} = { }_5 C_3 = 10.0 \][/tex]
The probability that all 3 chosen books are mysteries is inversely proportional to the number of ways to choose 3 mysteries, which can be expressed as:
[tex]\[ \text{Probability} = \frac{1}{10.0} = 0.1 \][/tex]
Hence, this statement is true.
### Statement 5:
"The probability that Mariah will not choose all mysteries can be expressed as [tex]\(1 - \frac{{ }_{2} C_3}{{ }_{20} C_3}\)[/tex]."
Here, [tex]\({ }_{2} C_3\)[/tex] denotes the number of ways to choose 3 books, all of which are not mysteries. Since [tex]\({ }_2 C_3 = 0\)[/tex] (because you can't choose 3 out of 2), the expression becomes:
[tex]\[ \frac{0}{20} = 0 \][/tex]
Thus, the probability that all 3 chosen books are not mysteries can be computed as:
[tex]\[ 1 - 0 = 1.0 \][/tex]
Hence, this statement is true.
### Conclusion:
All the given statements are true based on the provided numerical results.
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