During AMC testing, the AoPS Wiki is in read-only mode. No edits can be made.

# 2018 AMC 8 Problems/Problem 16

## Problem 16

Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?

$\textbf{(A) }1440\qquad\textbf{(B) }2880\qquad\textbf{(C) }5760\qquad\textbf{(D) }182,440\qquad \textbf{(E) }362,880$

## Solution 1

Since the Arabic books and Spanish books have to be kept together, we can treat them both as just one book. That means we're trying to find the number of ways you can arrange one Arabic book, one Spanish book, and three German books, which is just $5$ factorial. Now we multiply this product by $2!$ because there are $2$ factorial ways to arrange the Arabic books within themselves, and $4!$ ways to arrange the Spanish books within themselves. Multiplying all these together, we have $2!\cdot 4!\cdot 5!=\boxed{\textbf{(C) }5760}$.