Let's break down each statement one by one and analyze whether it is true based on the calculations:
1. There are [tex]\({ }_{20} C _3\)[/tex] possible ways to choose three books to read.
We need to find the total number of combinations of choosing 3 books from 20. The formula for combinations is given by [tex]\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)[/tex].
Here, we need to find [tex]\(\binom{20}{3}\)[/tex]. This evaluates to 1140.
Since the calculation aligns with the statement, this statement is true.
2. There are [tex]\({ }_5 C _3\)[/tex] possible ways to choose three mysteries to read.
The number of ways to choose 3 mysteries from 5 mysteries is given by [tex]\(\binom{5}{3}\)[/tex].
This evaluates to 10.
Since this is correct, the statement is true.
3. There are [tex]\({ }_{15} C _3\)[/tex] possible ways to choose three books that are not all mysteries.
Here, we need to find the number of combinations of choosing 3 books from the total 15 non-mystery books (7 biographies + 8 science fiction).
We calculate [tex]\(\binom{15}{3}\)[/tex]. This evaluates to 455.
The statement is confirmed to be true.
4. The probability that Mariah will choose 3 mysteries can be expressed as [tex]\(\frac{1}{\binom{5}{3}}\)[/tex].
To find the probability of choosing 3 mysteries, we divide the number of ways to choose 3 mysteries by the total number of ways to choose 3 books.
[tex]\(\text{Probability} = \frac{\binom{5}{3}}{\binom{20}{3}} = \frac{10}{1140} = 0.008771929824561403\)[/tex].
This does not align with [tex]\(\frac{1}{\binom{5}{3}}\)[/tex], which would be [tex]\(\frac{1}{10} = 0.1\)[/tex].
Therefore, this statement is false.
5. The probability that Mariah will not choose all mysteries can be expressed as [tex]\(1 - \frac{\binom{5}{3}}{\binom{20}{3}}\)[/tex].
First, we calculate the probability of choosing all mysteries, which is [tex]\(\frac{\binom{5}{3}}{\binom{20}{3}} = \frac{10}{1140} = 0.008771929824561403\)[/tex].
Then, the probability of not choosing all mysteries is:
[tex]\[
1 - \text{Probability of choosing all mysteries} = 1 - 0.008771929824561403 = 0.9912280701754386
\][/tex]
Since this matches our results, the statement is true.
Summarizing, the statements:
1, 2, 3, and 5 are true.
Statement 4 is false.
