Let's solve the problem step-by-step:
1. Define the total number of books:
The private library has a total of 400 books.
2. Define the difference between fiction and nonfiction books:
There are 40 more fiction books than nonfiction books.
3. Set up the equations:
Let \( x \) represent the number of nonfiction books. Then, the number of fiction books will be \( x + 40 \).
4. Formulate the total books equation:
The total number of books is the sum of fiction and nonfiction books:
[tex]\[
x + (x + 40) = 400
\][/tex]
Simplifying this equation:
[tex]\[
2x + 40 = 400
\][/tex]
[tex]\[
2x = 360
\][/tex]
[tex]\[
x = 180
\][/tex]
Therefore, there are 180 nonfiction books and \( 180 + 40 = 220 \) fiction books.
5. Calculate the probability that both Audrey and Ryan pick nonfiction books:
- The probability that Audrey picks a nonfiction book is:
[tex]\[
\frac{\text{Number of nonfiction books}}{\text{Total number of books}} = \frac{180}{400}
\][/tex]
- After Audrey picks a nonfiction book, there will be \( 180 - 1 = 179 \) nonfiction books left out of \( 400 - 1 = 399 \) books remaining.
- The probability that Ryan then picks a nonfiction book from the remaining books is:
[tex]\[
\frac{\text{Remaining nonfiction books}}{\text{Remaining total books}} = \frac{179}{399}
\][/tex]
6. Combine the probabilities:
The combined probability that both pick nonfiction books can be calculated by multiplying the individual probabilities:
[tex]\[
\left(\frac{180}{400}\right) \times \left(\frac{179}{399}\right)
\][/tex]
Given the steps we followed, the correct answer is:
[tex]\[
\boxed{\frac{180 \times 179}{400 \times 399}}
\][/tex]
Thus, the correct answer is:
D. [tex]\(\frac{180 \times 179}{400 \times 399}\)[/tex]
