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The problem involves finding the number of possible serial numbers for a bill under different conditions. The serial number consists of one letter followed by nine digits, and then another letter.

Let's address each part:

### (a) Letters and digits cannot be repeated.
- There are 26 letters in the English alphabet.
- There are 10 possible digits (0-9).

For the first letter, you have 26 options. For the nine digits, since they cannot be repeated, you have 10 choices for the first digit, 9 choices for the second digit, and so on. Lastly, for the final letter, you have 25 choices (since the letter cannot be repeated).

Therefore, the total number of possible serial numbers for part (a) is:
\[
26 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 25
\]

### (b) Letters and digits can be repeated.
- For the first letter, you still have 26 options.
- For each of the nine digits, you have 10 options since repetition is allowed.
- For the final letter, you again have 26 options.

The total number of possible serial numbers for part (b) is:
\[
26 \times 10^9 \times 26
\]

### (c) The letters are non-repeated consonants, and the digits can be repeated.
- There are 21 consonants in the English alphabet (considering 'y' as a consonant).
- For the first letter, you have 21 options, and for the final letter, you have 20 options (since the letter cannot be repeated).
- For each of the nine digits, you still have 10 options (repetition is allowed).

The total number of possible serial numbers for part (c) is:
\[
21 \times 10^9 \times 20
\]

Would you like me to calculate each of these values or provide more details?

Here are 5 related questions to deepen your understanding:
1. What would the solution look like if the number of digits were different, say 8 or 10?
2. How would the answer change if only vowels were allowed for the letters?
3. What if the digits had to follow a certain pattern, like being in ascending order?
4. How would the number of possibilities change if uppercase and lowercase letters were distinct?
5. What if the digits were restricted to only odd numbers?

**Tip**: When working with combinations and permutations, ensure you clearly define whether repetition is allowed and any restrictions on the available set of items.

The serial number on a bill consists of a letter followed by nine digits and then a letter. How many different serial numbers are possible, given the following conditions? (a) Letters and digits cannot be repeated, (b) Letters and digits can be repeated, (c) The letters are non-repeated consonants and the digits can be repeated.

Learn how to calculate the number of possible serial numbers for a bill under various conditions, including repetition and non-repetition of letters and digits. This problem involves combinatorial principles and is suitable for students in grades 9-12.

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