1. **Problem statement:** We need to find the number of 6-letter passwords using distinct letters from the English alphabet where the first letter must be a vowel. 2. **Important information:** - English alphabet has 26 letters. - Vowels are A, E, I, O, U (5 vowels). - Password length is 6. - Letters must be distinct. - First letter must be a vowel. 3. **Step 1: Choose the first letter (vowel).** There are 5 choices for the first letter. 4. **Step 2: Choose the remaining 5 letters.** After choosing the first vowel, 25 letters remain (26 - 1). We need to choose 5 distinct letters from these 25. 5. **Step 3: Arrange the remaining 5 letters.** Since order matters, we use permutations. Number of ways to arrange 5 letters out of 25 is $$P(25,5) = \frac{25!}{(25-5)!} = \frac{25!}{20!}$$ 6. **Step 4: Calculate total number of passwords.** Total = (choices for first letter) \times (arrangements of remaining 5 letters) $$\text{Total} = 5 \times P(25,5) = 5 \times \frac{25!}{20!}$$ 7. **Step 5: Calculate numeric value.** $$P(25,5) = 25 \times 24 \times 23 \times 22 \times 21 = 6375600$$ So, $$\text{Total} = 5 \times 6375600 = 31878000$$ **Final answer:** There are 31,878,000 possible passwords.