Question

mariah is randomly choosing three books to read from the following: 5 mysteries, 7 biographies, and 8 science fiction novels. which of these statements are true? check all that apply. there are $_{20}c_3$ possible ways to choose three books to read. there are $_{5}c_3$ possible ways to choose three mysteries to read. there are $_{15}c_3$ possible ways to choose three books that are not all mysteries. the probability that mariah will choose 3 mysteries can be expressed as $\frac{1}{_5c_3}$. the probability that mariah will not choose all mysteries can be expressed as $1 - \frac{_5c_3}{_{20}c_3}$.

Explanation:

Step1: Total number of books

First, calculate the total number of books. There are 5 mysteries, 7 biographies, and 8 science - fiction novels. So the total number of books $n = 5+7 + 8=20$.
The number of ways to choose 3 books from 20 is given by the combination formula $_{n}C_{r}=\frac{n!}{r!(n - r)!}$, where $n = 20$ and $r = 3$. So the number of ways to choose 3 books from 20 is $_{20}C_{3}$. So the first statement "There are $_{20}C_{3}$ possible ways to choose three books to read" is true.

Step2: Choosing 3 mysteries

The number of mystery books is 5. The number of ways to choose 3 books from 5 mystery books is given by the combination formula with $n = 5$ and $r = 3$, i.e., $_{5}C_{3}$. So the statement "There are $_{5}C_{3}$ possible ways to choose three mysteries to read" is true.

Step3: Choosing 3 non - all - mystery books

The number of ways to choose 3 books that are not all mysteries is equal to the total number of ways to choose 3 books minus the number of ways to choose 3 mystery books. The total number of ways to choose 3 books is $_{20}C_{3}$, and the number of ways to choose 3 mystery books is $_{5}C_{3}$. So the number of ways to choose 3 non - all - mystery books is $_{20}C_{3}-_{5}C_{3}$, not $_{15}C_{3}$ (since $20 - 5=15$, but the formula for non - all - mystery is not $_{15}C_{3}$ because we are not restricted to a subset of 15 books in the sense of a simple subtraction of the number of mystery books. We have to consider all possible combinations except the all - mystery ones). So the statement "There are $_{15}C_{3}$ possible ways to choose three books that are not all mysteries" is false.

Step4: Probability of choosing 3 mysteries

The probability of an event is given by the number of favorable outcomes divided by the number of total outcomes. The number of favorable outcomes (choosing 3 mysteries) is $_{5}C_{3}$, and the number of total outcomes (choosing 3 books from 20) is $_{20}C_{3}$. So the probability $P=\frac{_{5}C_{3}}{_{20}C_{3}}$, not $\frac{1}{_{5}C_{3}}$. So the statement "The probability that Mariah will choose 3 mysteries can be expressed as $\frac{1}{_{5}C_{3}}$" is false.

Step5: Probability of not choosing all mysteries

The probability of not choosing all mysteries is equal to 1 minus the probability of choosing all mysteries. The probability of choosing all mysteries is $\frac{_{5}C_{3}}{_{20}C_{3}}$. So the probability of not choosing all mysteries is $1-\frac{_{5}C_{3}}{_{20}C_{3}}$. So the statement "The probability that Mariah will not choose all mysteries can be expressed as $1-\frac{_{5}C_{3}}{_{20}C_{3}}$" is true.

Answer:

• There are $_{20}C_{3}$ possible ways to choose three books to read.
• There are $_{5}C_{3}$ possible ways to choose three mysteries to read.
• The probability that Mariah will not choose all mysteries can be expressed as $1-\frac{_{5}C_{3}}{_{20}C_{3}}$.