The problem reads:

"How many arrangements of the word GRACIOUS both begin and end in a vowel?"

I tried counting A and I as "one unit" and then figuring out how many possibilities, then I and O as "one unit" and figuring out how many possibilites there, and then O and U as "one unit" and figuring out how many possibilities, then A and O... you get the idea and then adding them together. But not only do I NOT know whether or not this is even right, (60480 possibilities) I'm almost certain there has to be a better way to do this.