How many ways are there to arrange the letters in UNIVERSALLY so that the four vowels appear in two cluster of two consecutive letters with at least 2 consonants between the two clusters?
For this, we will look at the compliment of the statement. When doing this we have to look at when their are no consonants in between the cluster of vowels and when there is 1 consnant in between the vowel clusters.
CASE 1: No consonants in between vowel cluster
Since there is no consonant in between the two pair of vowel clusters, we can line up all 4 vowels. We are going to treat UIEA as one letter for right now. We have UIEANVRSLLY. Since UIEA is one letter, we have a total of 8 letters. The ways to arrange this is $\cfrac{8!}{2!1!1!1!1!1!1!}$. Now lets revisit our vowel cluster. The ways to rearrange UIEA is 4!. Therefore the amount of ways to arrange UNIVERSALLY with no consonants in between vowel cluster is 4! $\cfrac{8!}{2!}$
CASE 2: 1 consonant in between vowel cluster
This is where I am stuck. The fact that there is 2 L's is confusing me and I do not understand how to proceed
Any help would be appreciated
Thank you!