How many arrangements of SYSTEMATIC are there in which each S is followed by a vowel. Why is the answer $$\frac{12\times8!}{2!2!}$$ and not $$\frac{12\times8!}{2!}\;\;?$$ Are the cases for example SESY the same as SYSE ?
Thanks
 A: I assume that you are treating Y as a vowel here?
You have two S's, and must choose a vowel to put after each; your vowels are E, A, I, and (apparently) Y.  So, you choose two out of those, in one of $\binom{4}{2}$ ways, to match with the S's.  Now, you need to arrange the 8 symbols consisting of your two S/vowel combinations and the remaining 6 symbols.  Because the only repeated character at this point is a T (because each S has a different vowel after it!), this can be done in $\frac{8!}{2!}$ ways.
So, all in all, I see a total of
$$
\binom{4}{2}\cdot\frac{8!}{2!}=\frac{4!}{2!\cdot 2!}\cdot\frac{8!}{2!}=120,960
$$
different combinations.
(Note that this does match the answer you were given.)
A: No, the cases of SESY are NOT the same as SYSE.  The issue lies with the number of repeated letters in SYSTEMATIC.  You have a repeated S and a repeated T.  Thus, you must divide by $2!$ for the S's and again by $2!$ again for the T's.  
Think of it this way.  Let $T_1=T=T_2$.  Then $T_1SESYT_2$ is different from $T_2SESYT_1$ if T's are different.  However, since they both are equal to $TSESYT$, you need to divide by $2!$ to account for this repeated permutation.
