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 20C3 possible ways to choose three books to read.
- There are 5C3 possible ways to choose three mysteries to read.
- There are 15C3 possible ways to choose three books that are not all mysteries.
- The probability that Mariah will choose 3 mysteries can be expressed as \( \frac{1}{5C3} \).
Answer (1)
To solve this problem, we need to analyze the statements by considering the number of ways Mariah can choose the books from given categories using combinations.
Statement 1: There are ( 3 20 ) possible ways to choose three books to read.
Mariah has a total of 20 books to choose from: 5 mysteries, 7 biographies, and 8 science fiction novels.
To find the number of ways to choose 3 books from 20, we use the combination formula ( r n ) = r ! ( n − r )! n ! .
Applying this formula, ( 3 20 ) = 3 ! ( 20 − 3 )! 20 ! = 1140 .
So, this statement is true.
Statement 2: There are ( 3 5 ) possible ways to choose three mysteries to read.
Mariah has 5 mystery books, and we need to find the number of ways to choose 3 mysteries.
Using the formula, ( 3 5 ) = 3 ! ( 5 − 3 )! 5 ! = 10 .
This statement is true.
Statement 3: There are ( 3 15 ) possible ways to choose three books that are not all mysteries.
If all three books are not mysteries, they should be chosen from the remaining books: 7 biographies and 8 science fiction novels.
This gives us a total of 15 non-mystery books.
Using the combination formula, ( 3 15 ) = 3 ! ( 15 − 3 )! 15 ! = 455 .
This statement is true.
Statement 4: The probability that Mariah will choose 3 mysteries can be expressed as ( 3 5 ) 1 .
To find the probability, we consider the number of ways to choose 3 mysteries ( ( 3 5 ) = 10 ) divided by the total number of ways to choose 3 books from all 20 books ( ( 3 20 ) = 1140 ).
Probability = ( 3 20 ) ( 3 5 ) = 1140 10 = 114 1 .
This is not equal to ( 3 5 ) 1 .
Therefore, this statement is false.
Based on the analysis, the true statements are 1, 2, and 3.
