Question on simple counting methods using combinatorics

Question

Five of 'a' type books, three of 'b' type books and two of 'c' type books are to be arranged in a row on a shelf. Find the number of ways for the arrangement if the books are all different, and each subject is grouped together.

I understand that you have to use permutations to calculate this, and so far what I've figured is that it should be $5! * 3! * 2!$. Would someone mind explaining why this is so but also why I am wrong (as the answer is actually $5! * 3! * 3! * 2!$)

Answer 1

You forgot to count the $3!$ different arrangements of the 3 subjects.

The number $5!\cdot 3!\cdot 2!$ does not count the two following different arrangements: $$A_1 A_2A_3 A_4 A_5 B_1B_2B_3 C_1 C_2$$ and $$C_1 C_2 A_1 A_2A_3 A_4 A_5 B_1B_2B_3.$$

Answer 2

HINT:

$$\underbrace{\underbrace{a_1a_2a_3a_4a_5}_{5!}\mid\underbrace{b_1b_2b_3}_{3!}\mid\underbrace{c_1c_2}_{2!}}_{3!}$$