Subjects combinatorics

Arrangements Pretty C79Ee4

Step-by-step solutions with LaTeX - clean, fast, and student-friendly.

Use the AI math solver

1. **State the problem:** We want to find the number of unique ways to arrange the letters in the word PRETTY. 2. **Identify the letters and their frequencies:** The word PRETTY has 6 letters: P, R, E, T, T, Y. Note that the letter T appears twice. 3. **Formula for permutations with repeated elements:** When some elements repeat, the number of unique arrangements is given by: $$\frac{n!}{n_1! \times n_2! \times \cdots}$$ where $n$ is the total number of letters, and $n_1, n_2, \ldots$ are the frequencies of each repeated letter. 4. **Apply the formula:** Here, $n=6$ (letters in PRETTY), and the letter T repeats twice, so $n_1=2$. 5. **Calculate factorials:** $$6! = 720$$ $$2! = 2$$ 6. **Calculate the number of unique arrangements:** $$\frac{6!}{2!} = \frac{720}{2} = 360$$ **Final answer:** There are 360 unique ways to arrange the letters in the word PRETTY.