The chairs of an auditorium are to be labeled with an uppercase English letter followed by a positive integer not exceeding 100. What is the largest number of chairs that can be labeled differently?

English letter have 26 alphabets and a chair can be labelled by the following two ways.

1) Assigning to the seat one of the 26 uppercase letters

2)Assigning to the seat one of the 100 possible integers

using the product rule, total number of different ways of labelling a chair is $26*100=2600$

Therefore, the largest number of chairs that can be labeled differently is 2600.