1. **State the problem:** We want to find the number of unique ways to arrange the letters in the word HATTER. 2. **Identify the letters and their frequencies:** The word HATTER has 6 letters: H, A, T, T, E, R. - H appears 1 time - A appears 1 time - T appears 2 times - E appears 1 time - R appears 1 time 3. **Formula for permutations of letters with repetitions:** $$\text{Number of unique arrangements} = \frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}$$ where $n$ is the total number of letters, and $n_1, n_2, ..., n_k$ are the frequencies of each repeated letter. 4. **Apply the formula:** Total letters $n = 6$ Repeated letters: T appears twice, so $n_T = 2$ $$\text{Number of unique arrangements} = \frac{6!}{2!}$$ 5. **Calculate factorials:** $$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$$ $$2! = 2 \times 1 = 2$$ 6. **Simplify the expression:** $$\frac{6!}{2!} = \frac{720}{2} = 360$$ 7. **Final answer:** There are **360** unique ways to arrange the letters in the word HATTER.