Thread: Need help with a permutation problem...

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.

Let's see if I count all the ways.

There are 4 vowels in the word and they can be placed one on either end in

P(4,2)=12 ways. Place two of the vowels and arrange the other 6 letters in 6!=720 ways.

720*12=8640 ways to arrange GRACIOUS with a vowel on the ends.

Ah, so just figure out the number of possibilities for vowels and then the consonants? Thanks.