3. The letters of the word MATHEMATICS are rearranged to form distinct strings of the same 11 letters. What proportion of these strings do not contain the string MATH?
Answer: $\frac{491}{495}$
Solution: By treating the string MATH as a single character, we find that there are exactly eight distinct "letters" to rearrange. Hence, there are 8 ! such words out of a total of $\frac{11!}{2!2!2!}$. This makes for a proportion of $\frac{8!\cdot 8}{11!}=\frac{4}{495}$, and so $\frac{491}{495}$ of the words do not contain the aforementioned string.