Consider the integers from 0 to 999, written in the normal way. We will ask several questions about these numbers, and then ask you some questions about the questions.
a) How many of these numbers contain exactly one digit 0?
b) How many of these numbers contain at least one digit 0?
c) What is the total number of digits 0 that appear in all these numbers?
Now, sometimes the numbers from 0 to 999 have to be written with three digits no matter what, for instance, in marking answers on a computer scan sheet for the AIME exam. In such cases, 0 is written as 000 and 23 is written as 023. Answer a)-c) again for the numbers from 000 to 999 written this way. That is, answer
a) How many of these specially written numbers contain exactly one digit 0?
b) How many of these specially written numbers contain at least one digit 0?
c) Which set of questions, a)-c) or a′)-c′), did you prefer? Why?
d) Of your six solutions, which did you like best? Why?