Combinatorics, arrangements (edited) "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. 
 A: Your argument is fine (as previously noted), and there's probably no significantly better approach to counting them.
If you want to double-check your result, here's some GAP code which can list all the possibilities.
A:=["S","L","U","M","G","U","L","L","I","O","N"];
T:=Arrangements(A,Size(A));
count:=0;

for P in T do

  # where the last L is
  a:=Maximum(Positions(P,"L"));

  # where the first non-L consonant is
  b:=Minimum(Position(P,"S"),Position(P,"M"),Position(P,"G"),Position(P,"N"));

  if(b>a) then
    count:=count+1;
    Print(P,"\n");
  fi;

od;

Print(count,"\n");

and it found 95040, matching your result $4! \frac{11!}{7!\ 2!}$.
A: 
Here is an automaton that will generate the required words.
A good word has three parts : A is before the third L, then the third L, then the suffix B that  contains no L's.
the equations are :
$A = 1 + (l+i+o+u).A$
$B = 1 + (g + m + n + s + i + o + u) .B$
$W = A.l.B$
the generating function for W is
${1 \over 1- (l+i+o+u)} .l. {1 \over 1- (g+m+n+s + i + o + u)}$
and we are interested in the coefficient of 
$l^3.g.m.n.s.u^2.i.o$ which is $95040$.
The third L is necessary and it may be find (hidden) both in solution and in algorithm.
As one may see, such a problem involves only the sum and the product rules, and some coefficient stuff.
But, since 11 = (1+1+2) + (3+1+1+1+1) and there are four types of letters I can not imagine a shorter answer than the first, which is a product of four (factorial) factors.
U-type : 2 letters
I-type : 2 letters
S-type : 4 letters, consonants
L type : 3 letter, a multiple L
we have: 
$11= 2+2+3+4, $
$7 = 3+4,$ 
$4 = 4, $
$2 = 2$.
