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.

  • 1
    $\begingroup$ possible duplicate of In how many ways can the word ELEEMOSYNARY be arranged. $\endgroup$ – Travis Sep 30 '14 at 13:00
  • $\begingroup$ It is similar, but different. $\endgroup$ – user247 Sep 30 '14 at 13:12
  • $\begingroup$ Ah, I see now that there's a condition on the positioning of the $L$'s. This is indeed different (sorry, some variation of this problem, usually without such a modification, seems to be posted daily, and I glossed over the first sentence), and I've retracted my vote to close. $\endgroup$ – Travis Sep 30 '14 at 13:32
  • $\begingroup$ The $3!$ for the L's isn't needed, since the L's are identical. You also need to change the factor of 3, since you are not taking into account that the vowels can be interspersed between the other letters. $\endgroup$ – user84413 Sep 30 '14 at 19:34

First we make a $7$-letter sequence using only consonants, with the L's in front. There are $4!$ ways to do this. Leave fat gaps between successive consonants, we may want to slip vowels between them.

The I can be placed in our pure consonant sequence in $8$ ways.

Then the O can be inserted in the resulting sequence in $9$ ways.

The U's are more tricky. Either we use UU, which can be placed in $10$ ways, or we choose $2$ of the $10$ "gaps" in the $9$-letter sequence we have so far, to slip a U into. So the U's can be placed in $10+\binom{10}{2}$ ways.


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.


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