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 $\frac{1}{_{5}c_{3}}$. the probability that mariah will not choose all mysteries can be expressed as $1 - \frac{_{5}c_{3}}{_{20}c_{3}}$.

Explanation:

Step1: Calculate total number of books

There are \(5 + 7+8=20\) books. 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.

Step2: Calculate ways to choose 3 mysteries

There are 5 mysteries. The number of ways to choose 3 mysteries out of 5 is \(_{5}C_{3}\).

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 \(_{15}C_{3}\).

Step4: Calculate probability of choosing 3 mysteries

The probability of choosing 3 mysteries is \(\frac{_{5}C_{3}}{_{20}C_{3}}\), not \(\frac{1}{_{5}C_{3}}\).

Step5: Calculate probability of not 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 \(1-\frac{_{5}C_{3}}{_{20}C_{3}}\).

Answer:

There are \(_{20}C_{3}\) possible ways to choose three books to read. - True
There are \(_{5}C_{3}\) possible ways to choose three mysteries to read. - True
There are \(_{15}C_{3}\) possible ways to choose three books that are not all mysteries. - False
The probability that Mariah will choose 3 mysteries can be expressed as \(\frac{1}{_{5}C_{3}}\). - False
The probability that Mariah will not choose all mysteries can be expressed as \(1 - \frac{_{5}C_{3}}{_{20}C_{3}}\). - True

So the correct statements are:

• 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 \(1 - \frac{_{5}C_{3}}{_{20}C_{3}}\).