1. **Problem Statement:** We need to find the number of arrangements of the word "ABSOLUTE" (8 distinct letters) such that the word starts with a vowel (A, E, O, or U) and ends with either "A" or "T". 2. **Identify vowels and constraints:** The vowels available for the first position are A, E, O, U, so there are 4 choices. 3. **Last letter choices:** The last letter can be either "A" or "T", so there are 2 choices. 4. **Consider overlap cases:** Since "A" is both a vowel and an option for the last letter, we must consider cases where the first and last letters might be the same. 5. **Case 1: First letter is A, last letter is A:** - First position: A (1 way) - Last position: A (1 way) - Remaining letters to arrange: 6 letters (B, S, O, L, U, T) - Number of arrangements for middle 6 letters: $6! = 720$ 6. **Case 2: First letter is A, last letter is T:** - First position: A (1 way) - Last position: T (1 way) - Remaining letters: 6 letters (B, S, O, L, U, E) - Number of arrangements: $6! = 720$ 7. **Case 3: First letter is E, O, or U (3 choices), last letter is A:** - First position: 3 ways - Last position: A (1 way) - Remaining letters: 6 letters (B, S, L, T, plus the two vowels not chosen for first position) - Number of arrangements: $6! = 720$ 8. **Case 4: First letter is E, O, or U (3 choices), last letter is T:** - First position: 3 ways - Last position: T (1 way) - Remaining letters: 6 letters (B, S, L, A, plus the two vowels not chosen for first position) - Number of arrangements: $6! = 720$ 9. **Calculate total arrangements:** $$\text{Total} = (1 \times 1 \times 720) + (1 \times 1 \times 720) + (3 \times 1 \times 720) + (3 \times 1 \times 720) = 720 + 720 + 2160 + 2160 = 5760$$ 10. **Final answer:** There are $\boxed{5760}$ different arrangements where the word starts with a vowel and ends with "A" or "T".