To solve this problem step-by-step, let's first determine the number of fiction and nonfiction books in the library.
1. We know that there are a total of 400 books.
2. Let the number of nonfiction books be [tex]\( N \)[/tex].
3. Since there are 40 more fiction books than nonfiction books, the number of fiction books is [tex]\( N + 40 \)[/tex].
From the equation for the total number of books:
[tex]\[ N + (N + 40) = 400 \][/tex]
4. Simplify and solve for [tex]\( N \)[/tex]:
[tex]\[ 2N + 40 = 400 \][/tex]
[tex]\[ 2N = 360 \][/tex]
[tex]\[ N = 180 \][/tex]
So, there are [tex]\( 180 \)[/tex] nonfiction books and:
[tex]\[ N + 40 = 180 + 40 = 220 \][/tex]
220 fiction books.
Next, we need to find the probability that both Audrey and Ryan pick nonfiction books.
5. The probability that Audrey picks a nonfiction book:
[tex]\[ P(\text{Audrey picks nonfiction}) = \frac{180}{400} \][/tex]
6. After Audrey picks a nonfiction book, there are [tex]\( 179 \)[/tex] nonfiction books and [tex]\( 399 \)[/tex] books remaining. So, the probability that Ryan picks a nonfiction book after Audrey has already picked one:
[tex]\[ P(\text{Ryan picks nonfiction}) = \frac{179}{399} \][/tex]
7. The combined probability that both Audrey and Ryan pick nonfiction books:
[tex]\[ P(\text{both pick nonfiction}) = P(\text{Audrey picks nonfiction}) \times P(\text{Ryan picks nonfiction}) \][/tex]
[tex]\[ P(\text{both pick nonfiction}) = \frac{180}{400} \times \frac{179}{399} \][/tex]
Thus, the probability that both Audrey and Ryan pick nonfiction books is:
[tex]\[ \frac{180 \times 179}{400 \times 399} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{180 \times 179}{400 \times 399}} \][/tex]
So, the correct option is:
[tex]\[ \text{option B} \][/tex]
