We are tasked with determining how many unique student numbers are possible under the given constraints. The student number consists of 5 characters: the first 4 are digits, and the last character is a letter.

Step-by-Step Breakdown:

1. First 4 characters (digits):

   • The digits must be selected from 0 to 9, and no digits can repeat.
   • For the first digit, we have 10 possible choices (0 to 9).
   • For the second digit, since one digit has already been used, we have 9 choices.
   • For the third digit, 8 choices remain.
   • For the fourth digit, we have 7 choices left.

   Therefore, the total number of ways to choose the first four digits is: $10 \times 9 \times 8 \times 7$

2. Last character (letter):

   • The last character must be a letter, and there are 26 letters in the alphabet (assuming we are using English alphabet).

   So, there are 26 choices for the last character.

Total number of unique student numbers:

To find the total number of possible student numbers, we multiply the number of possibilities for the digits by the number of possibilities for the letter: $10 \times 9 \times 8 \times 7 \times 26$

Now, let's compute this.

$10 \times 9 = 90$ $90 \times 8 = 720$ $720 \times 7 = 5040$ $5040 \times 26 = 131,040$

Thus, the total number of unique student numbers possible is: $\boxed{131,040}$

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. What if the digits could repeat? How would that change the solution?
2. What if the last character could also be a digit or a letter? How many combinations would be possible then?
3. What is the probability of randomly generating a valid student number?
4. How many valid numbers would be possible if we restricted the digits to not start with zero?
5. How would the total number change if the last two characters had to be letters?

Tip: When calculating permutations, remember that if repetition is not allowed, the number of available choices decreases with each selection.