jmc

If every letter of ``Hawaii" were different, there would be $6!=6\cdot 5\cdots 2\cdot 1$ distinct six-letter rearrangements, since for the first letter of the rearrangement there would be six letters to choose from, for the second there would be five, etc. However, ``Hawaii" contains two copies of the letter $a$ and two copies of the letter $i$. Thus, we must divide by $2$ to eliminate the overcounting we get from two indistinguishable $a$'s, and we must divide by $2$ again to eliminate the overcounting we get from two indistinguishable $i$'s. Our final count is therefore $\frac{6!}{2\cdot 2}$. Canceling the $4$ top and bottom gives $6\cdot 5\cdot 3\cdot 2=30\cdot 6=\boxed{180}$.