1. **State the problem:** We want to find the number of distinguishable ways to arrange the letters in the word "statistics". 2. **Formula used:** The number of distinguishable permutations of a word with repeated letters is given by: $$\frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}$$ where $n$ is the total number of letters, and $n_1, n_2, \ldots, n_k$ are the frequencies of each repeated letter. 3. **Count the letters in "statistics":** - Total letters $n = 10$ - Letter counts: - $s$: 3 times - $t$: 3 times - $a$: 1 time - $i$: 2 times - $c$: 1 time 4. **Apply the formula:** $$\frac{10!}{3! \times 3! \times 2! \times 1! \times 1!} = \frac{10!}{3! \times 3! \times 2!}$$ 5. **Calculate factorials:** - $10! = 3628800$ - $3! = 6$ - $2! = 2$ 6. **Simplify the denominator:** $$3! \times 3! \times 2! = 6 \times 6 \times 2 = 72$$ 7. **Calculate the number of distinguishable arrangements:** $$\frac{3628800}{72} = 50400$$ **Final answer:** There are **50400** distinguishable ways to arrange the letters in "statistics".