Where Have I Gone Wrong? 2 Combinatorics Problems

Question

Three physics books, five biology books, a dictionary and two comic books are stored on a bookshelf.

(a) Determine the number of possible arrangements where the two comic books are not next to each other.

(b) If four books are selected randomly from the book shelf, what is the probability that exactly two physics books are selected?

I have an answer to both questions, but they are both wrong. I'm trying to figure out why but I can't.

For (a) I put $11!-10!$. Why is this wrong?

For (b) I put $3/11 \cdot 2/10 \cdot 8/9 \cdot 7/8$, but the answer is $1/110$. Why is that wrong?

Thanks in advance.

Answer 1

• (a): There are $10!\cdot \color{blue}{2!}$ arrangements with the comics beside each other. So, you need to subtract a bit more.
• (b): Since order does not matter, you get $\frac{\binom 3 2 \cdot \binom 8 2}{\binom{11}{4}}$. So, the given solution is wrong.