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"How many ways can the letters in the word SLUMGULLION be arranged so that the three L’s precede all the other consonants?"

My work is below: Can someone also solve this ONLY using the multiplication rule, permutations, and permutations with repetitions?

We have 3 L's and the other 4 consonants are S,M,G,N. That is, our consonants are LLLSMGN, call them all X for the moment. Then we have XXXXXXXUUIO. The number of arrangements of these letters is $\frac{11!}{7!2!}$. Hence the answer is $4!*\frac{11!}{7!2!}$ since there are $4!$ ways to arrange the 4 consonants other than the L's.

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    Hmm, I got the same answer as you. – 2012-03-26
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    Your reasoning is correct. I also solved the problem a different way (using combinations) and got the same answer. – 2012-03-26

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