I'm told that there are five distinct books of category $A$, three distinct books of category $B$, and two of category $C$.
I'm then asked the following question:
In how many ways can these books be arranged on a shelf if all five category $A$ books are on the left, and both category $C$ books are on the right?
In what follows is my attempt:
$$\underset{\_}{5}\underset{\_}{4}\underset{\_}{3}\underset{\_}{2}\underset{\_}{1}\underset{\_}{3}\underset{\_}{2}\underset{\_}{1}\underset{\_}{2}\underset{\_}{1}$$
Best Answer
That is correct. The categories must be arranged as follows:
$$ABC$$
Within $A$, there are $5!$ possible arrangements, within $B$ there are $3!$, and within $C$ there are $2!$. Using the rule of product, the total number of configurations allowable on the bookshelf is:
$$5!\cdot3!\cdot2!$$