Problem of the Week Problem E Red Dog

We can take any word and rearrange all the letters to get another “word”. These new “words” may be nonsensical. For example, you can rearrange the letters in $$MATH$$ to get $$MTHA$$.

Nalan wants to rearrange all the letters in $$RED DOG$$. However, she uses the following rules:

• the letters $$R$$, $$E$$, and $$D$$ cannot be adjacent to each other and in that order, and

• the letters $$D$$, $$O$$, and $$G$$ cannot be adjacent to each other and in that order.

For example, the “words” $$DOGRED$$, $$DDOGRE$$, $$GDREDO$$, and $$DREDOG$$ are examples of unacceptable words in this problem, but $$DROEGD$$ is acceptable.

How many different arrangements of the letters in $$RED DOG$$ can Nalan make if she follows these rules?