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 _20c_3 possible ways to choose three books to read. there are _5c_3 possible ways to choose three mysteries to read. there are _15c_3 possible ways to choose three books that are not all mysteries. the probability that mariah will choose 3 mysteries can be expressed as 1 / _5c_3. the probability that mariah will not choose all mysteries can be expressed as 1 - _5c_3 / _20c_3.

Explanation:

Step1: Calculate total number of books

Total number of books is $5 + 7+8=20$. The number of ways to choose 3 books out of 20 is given by the combination formula $_{n}C_{r}=\frac{n!}{r!(n - r)!}$, so there are $_{20}C_{3}$ ways to choose 3 books. This statement is true.

Step2: Calculate ways to choose 3 mysteries

There are 5 mysteries. The number of ways to choose 3 mysteries out of 5 is $_{5}C_{3}$. This statement is true.

Step3: Calculate ways to choose non - all - mysteries

The number of ways to choose 3 books that are not all mysteries is the total number of ways to choose 3 books minus the number of ways to choose 3 mysteries. The total number of ways to choose 3 books is $_{20}C_{3}$ and the number of ways to choose 3 mysteries is $_{5}C_{3}$, not $_{15}C_{3}$. This statement is false.

Step4: Calculate probability of choosing 3 mysteries

The probability of choosing 3 mysteries is $\frac{_{5}C_{3}}{_{20}C_{3}}$, not $\frac{1}{_{5}C_{3}}$. This statement is false.

Step5: Calculate probability of not choosing all mysteries

The probability of choosing all mysteries is $\frac{_{5}C_{3}}{_{20}C_{3}}$. The probability of not choosing all mysteries is $1-\frac{_{5}C_{3}}{_{20}C_{3}}$. This statement 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 $1-\frac{_{5}C_{3}}{_{20}C_{3}}$.