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Problem 368

Combinatorics Difficulty 2.5 Multiple choice CEMC Pascal · Canada · 2016

An integer from 10 to 99 inclusive is randomly chosen so that each such integer is equally likely to be chosen. The probability that at least one digit of the chosen integer is a 6 is

Pick one

Official solution

From 10 to 99 inclusive, there is a total of 90 integers. (Note that $90 = 99 - 10 + 1$ .)

If an integer in this range includes the digit 6, this digit is either the ones (units) digit or the tens digit.

The integers in this range with a ones (units) digit of 6 are 16, 26, 36, 46, 56, 66, 76, 86, 96.

The integers in this range with a tens digit of 6 are 60, 61, 62, 63, 64, 65, 66, 67, 68, 69.

In total, there are 18 such integers. (Notice that 66 is in both lists and $9+10-1=18$ .)

Therefore, the probability that a randomly chosen integer from 10 to 99 inclusive includes the digit 6 is $\frac{18}{90} = \frac{1}{5}$ .

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