Suppose that I have $6$ different books, $2$ of which are math books. In how many ways can I stack my $6$ books on a shelf if I do not want the math books to be next to each other?
One way to approach this is to first fix the positions of the non-math books, and then see how many ways I can place the math books in the remaining spots. Since I have $4$ non-math books, there are $4! = 24$ ways to arrange them in a row. This can be done in ${5 \choose 2} = 10$ ways. Finally, once I have chosen the gaps, there are $2! = 2$ ways to arrange the math books in them. Therefore, by the multiplication principle, the total number of ways to stack the books is $24 \times 10 \times 2 = 480$. 