To solve this problem, we need to use combinatorics to determine the total number of ways to select 4 books from the 12 books and then calculate the probability of selecting 2 novels and 2 biographies.

Step 1: Calculate the Total Number of Ways to Select 4 Books

The total number of ways to choose 4 books out of 12 is given by the combination formula $\binom{n}{k}$, which represents the number of ways to choose $k$ items from $n$ items without regard to order:

$\binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!}$

Step 2: Calculate the Number of Ways to Select 2 Novels and 2 Biographies

The number of ways to choose 2 novels out of 7 is:

$\binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7!}{2! \cdot 5!}$

The number of ways to choose 2 biographies out of 5 is:

$\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!}$

Step 3: Calculate the Probability

The probability of selecting 2 novels and 2 biographies is the ratio of the number of favorable outcomes to the total number of outcomes.

Number of Favorable Outcomes

First, calculate $\binom{7}{2}$ and $\binom{5}{2}$:

$\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21$

$\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10$

So, the number of ways to select 2 novels and 2 biographies is:

$\binom{7}{2} \times \binom{5}{2} = 21 \times 10 = 210$

Total Number of Outcomes

Next, calculate $\binom{12}{4}$:

$\binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = 495$

Finally, the probability of selecting 2 novels and 2 biographies is:

$P(\text{2 novels and 2 biographies}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{210}{495} = \frac{42}{99} \approx 0.4242$

Final Answers

1. The number of different groups of 4 books that Dan can select is $495$.
2. The probability that he randomly selects 2 novels and 2 biographies is $\frac{42}{99} \approx 0.4242$.

Would you like more details or have any questions?

Here are 8 related questions for further practice:

1. How many ways can Dan choose 3 novels and 1 biography?
2. What is the probability that Dan randomly selects all novels?
3. How many ways can Dan choose 1 novel and 3 biographies?
4. What is the probability that Dan selects at least one biography?
5. How many ways can Dan choose 4 biographies?
6. What is the probability that Dan randomly selects all biographies?
7. If Dan selects 4 books randomly, what is the probability that none of them are novels?
8. How many ways can Dan choose 2 novels and 1 biography from the remaining books if he has already chosen 1 book?

Tip: When working with combinations, ensure you correctly identify the total number of items and the number of selections to avoid calculation errors.