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Problem 444

Combinatorics Difficulty 2.7 Multiple choice CEMC Gauss (Grade 7) · Canada · 2016

A 10 by 10 grid is created using 100 points, as shown.

Point $P$ is given. One of the other 99 points is randomly chosen to be $Q$ . What is the probability that the line segment $PQ$ is vertical or horizontal?

Pick one

Official solution

Line segment $PQ$ is vertical if $Q$ is chosen from the points in the column in which $P$ lies.

This column contains 9 points other than $P$ which could be chosen to be $Q$ so that $PQ$ is vertical.

Line segment $PQ$ is horizontal if $Q$ is chosen from the points in the row in which $P$ lies.

This row contains 9 points other than $P$ which could be chosen to be $Q$ so that $PQ$ is horizontal.

Each of these 9 points is different from the 9 points in the column containing $P$ .

Thus, there are $9+9=18$ points which may be chosen to be $Q$ so that $PQ$ is vertical or horizontal.

Since there are a total of 99 points to choose $Q$ from, the probability that $Q$ is chosen so that $PQ$ is vertical or horizontal is $\frac{18}{99}$ or $\frac{2}{11}$ .

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