How many ways can the letters in the word SLUMGULLION be arranged so that the three L's precede all the other consonants.
Attempt: There are 11 letters, and there are 3 Ls, 4 vowels: U U I O, and 4 consonants: S M G N. Then Ls can be arranged in 3!, vowels in 4!/2!, and consonants in 4! ways. Let V = vowel, L = L, and C = consonant. The number of ways of for L to be before all other consonants are the possible combinantions VLC, LVC, LCV. Thus there are 3. Then we multiply 3(3!*4! *4!/[2!]) is this correct? Thank you for any feedback.