In this problem, we must check if the number of all possible variations of the five-digit ID code will be greater than the expected 75,000 students.
If all the numeric symbols from 0 to 9 will be used in the ID, there are ten possible values for each of the five digits. Therefore, the total number of possibilities (N) can be determined as follows:
Since the total number of possible five-digit IDs is bigger than the expected number of students, this system will be enough.
The reason for this answer is algebra. Since you give a full explanantion on it I'm not sure if I answered it correctly. But here is what I think from what you gave of the question.