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Problem 153

Combinatorics Difficulty 1.4 Multiple choice CEMC Fermat · Canada · 2015

Three-digit positive integers such as 789 and 998 use no digits other than 7, 8 and 9. In total, how many three-digit positive integers use no digits other than 7, 8 and 9?

Pick one

Official solution

For each of the three digits of such a positive integer, there are three choices of digit (7, 8 or 9).

Therefore, there are $3\cdot 3 \cdot 3 = 27$ possible integers that use no digits other than 7, 8 or 9.

(We note that there are 9 such integers beginning with each of 7, 8 and 9.

The 9 such integers beginning with 7 are 777, 778, 779, 787, 788, 789, 797, 798, 799.)

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.