EDIT: I was using the spelling originally given in the title, rather than in the statement of the problem; so I have edited my answer. There are $\displaystyle\frac{11!}{3!2!}$ total ways to arrange the letters of SLUMGULLION, and there are $\displaystyle\frac{7!}{3!}=840$ ways to arrange the consonants L,L,L,S,M,G,N. $\;\;$ There are $4!$ of these arrangements with the L's in front of the other consonants (since we have LLL_ _ _ _ with $4!$ ways to arrange S,M,G,N) so there are $\displaystyle\frac{4!}{840}\left(\frac{11!}{3!2!}\right)=95,040$ possible ways to do this. Alternatively, first arrange the 7 consonants L,L,L,S,M,G,N in order with the L's in front; as above, there are $4!$ ways to do this. Next we can place 4 spaces for the vowels between these letters, so there are $\dbinom{11}{4}$ ways to do this ${\hspace.3 in}$(since there are 4 spaces and 7 dividers). Finally, we can arrange the vowels U,U,I,O in these spaces in $\displaystyle\frac{4!}{2!}=12$ ways; so we have $4!\dbinom{11}{4}\cdot12=95,040$ possible arrangements.
Discussion :: Permutation and Combination - General Questions (Q.No.2)
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