Problem:
Eight points are spaced at intervals of one unit around a square, as shown. Two of the points are chosen at random. What is the probability that the points are one unit apart?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Choose two points. Any of the points can be the first choice, and any of the other points can be the second choice. So there are
ways of choosing the points in order. But each pair of points is counted twice, so there are possible pairs.
Label the eight points as shown. Only segments , and are unit long. So of the possible segments are unit long, and the probability that the points are one unit apart is .
Pick the two points, one at a time. No matter how the first point is chosen, exactly of the remaining points are unit from this point. So the probability of the second point being unit from the first is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions