How many arrangements of the word CHARACTERISTIC are there?
I'm assuming you mean if you use all of the letters.
If I counted correctly, there are 3 C's, 1 H, 2 A's, 2 R's, 2 T's, 1 E, 2 I's, and 1 S.
then the number of distinguisable permutations of the 14 letters of "characteristic" is
$\displaystyle \binom{14}{3} \binom{11}{1} \binom{10}{2} \binom{8}{2} \binom{6}{2} \binom{4}{1} \binom{3}{2} \binom{1}{1} = \frac{14!}{3!1!2!2!2!1!2!1!}$