1. **Problem statement:** Calculate the number of different serial numbers possible on a dollar bill where the serial number consists of a letter, followed by eight digits, and then a letter. 2. **Given:** - Number of letters in the alphabet = 26 - Number of vowels = 5 (not directly needed here) - Number of consonants = 21 (not directly needed here) - Digits = 0 to 9 (10 digits) 3. **Part 1: Letters and digits can be repeated.** - The serial number format is: Letter (1) + Digit (8) + Letter (1) - Number of choices for each letter = 26 - Number of choices for each digit = 10 4. **Formula:** $$\text{Total possibilities} = (\text{letters}) \times (\text{digits})^8 \times (\text{letters})$$ 5. **Calculation:** $$26 \times 10^8 \times 26 = 26 \times 26 \times 10^8 = 676 \times 10^8 = 67600000000$$ 6. **Part 2: Letters and digits cannot be repeated.** - Letters: 26 letters, no repetition, so first letter has 26 choices, last letter has 25 choices (since one letter used already) - Digits: 10 digits, no repetition, 8 digits chosen in order without repetition - Number of ways to choose 8 digits without repetition from 10 digits in order is a permutation: $$P(10,8) = \frac{10!}{(10-8)!} = \frac{10!}{2!}$$ 7. **Formula:** $$\text{Total possibilities} = 26 \times P(10,8) \times 25 = 26 \times \frac{10!}{2!} \times 25$$ 8. **Calculate permutations:** $$10! = 3628800$$ $$2! = 2$$ $$P(10,8) = \frac{3628800}{2} = 1814400$$ 9. **Final calculation:** $$26 \times 1814400 \times 25 = 26 \times 25 \times 1814400$$ $$= 650 \times 1814400 = 1179360000$$ **Answer:** - Part 1: $67600000000$ - Part 2: $1179360000$