Problem:
A collection of eight different books consists of two books on artificial intelligence, three books on operating systems, and three books on data structures.
i) How many ways can the books arranged on a shelf so that all books on a single subject are together?
ii) How many ways can the books arranged on a shelf so that the three books on operating systems are together?
iii) How many ways can the books be arranged on a shelf so that the two books on artificial intelligence occur at the right end of the arrangement?
My Attempt:
Let AI = A, OS = B, DS = C.
Possible Arrangements:
AABBBCCC AACCCBBB BBBAACCC BBBCCCAA CCCAABBB CCCBBBAA
Therefore, answer is six for part (i). However, the correct answer is meant to be 432. Where did I go wrong?
I think that the books should be considered different (even if they are about the same subject). So for part (i) we have $3!$ ways to arrange the subjects, $2!$ to arrange the AI books, $3!$ for OS books, and $3!$ for C books. So the answer should be $$3!\cdot 2!\cdot 3!\cdot 3!$$ As regards (ii), we can place the "block" of $3$ books on OS in $(8-3+1)\cdot 3!$ and then we fill the remaining $(8-3)$ places in $(8-3)!$ getting $$(8-3+1)\cdot 3!\cdot (8-3)!$$ Now are you able to answer to (iii)?