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Problem 831

AMC 12 late, AIME early

Combinatorics Difficulty 4.5 Multiple choice Progetto Olimpiadi di Matematica GARA di SECONDO LIVELLO · Italy

Last New Year's Eve, glasses shaped like the number "2009" with lenses in place of the two zeros were very much in fashion. In order to make similar glasses, it is necessary that the number representing the year contain two or more consecutive zeros (for example 3500 works, 2010 does not). How many years between the year 999 and the year 9999 contain two or more consecutive zeros in their writing?

Pick one

Official solution

Solution:

Note that if instead we had counted the number of ways in which one can substitute a digit from 1 to 9 for $X$ and a digit from 0 to 9 for $Y$ in the two patterns $X00Y$ and $XY00$ , we would have obtained an incorrect result because we would have counted twice the years of the form $X000$ .

The problem can be solved with an almost identical reasoning also using the inclusion-exclusion principle.

Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty, ordering) added by this project.