Professor Shortcut records his grades using only his student's first and last initials. What is the smallest class size that will definitely force Professor Shortcut to use a different system?

To determine the smallest class size that will definitely force Professor Shortcut to use a different grading system, we need to consider the number of unique combinations of first and last initials. Assuming that the first and last initials are represented by uppercase letters of the English alphabet (A-Z), we have 26 choices for both the first and last initials. The total number of unique combinations can be calculated by multiplying the number of choices for the first initial by the number of choices for the last initial. Therefore, the smallest class size that will definitely force Professor Shortcut to use a different system is when the number of unique combinations exceeds the number of students in the class. Let's calculate the smallest class size: Number of choices for first initial = 26 Number of choices for last initial = 26 Total number of unique combinations = 26 * 26 = 676 Therefore, the smallest class size that will definitely force Professor Shortcut to use a different system is 677 students. Once the class size reaches 677 or more, it will be necessary for Professor Shortcut to use a different system to differentiate between the students.