Official Guide Explanation:
Problem Solving #116




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Solution and Metadata

Question: 116
Page: 168
Difficulty: 6 (Moderately Difficult)
Category 1: Word Problems > Data Interpretation >
Category 2: Algebra > Linear Equations-One Unk >

Explanation: Without doing much math, you can easily deduce the correct choice. It's clear from the diagram that there are dots in fewer than half of the boxes. If the table were extended to 30 cities, there would be 30(30) = 900 boxes, so if the same rule held, there would be fewer than (900/2) = 450 dots. Since choice (A), 60, is far too small, the only possible choice is (B), 435.

More rigorously, this is a combinations problem. There is one dot per pair of cities, so the question could be rephrased as: "In a group of 30 cities, how many distinct pairs of cities are there?" That's a textbook combinations problem, so use the relevant equation: ((n!)/(k!(n - k)!)), where n is the population (in this case, 30) and k is the number that will be selected (in this case, 2). Plugging in: ((30!)/(2!(28!))). It would be foolish to try to find 30!, so recognize that 30! = (30)(29)(28!) and cancel out 28! from the numerator and denominator. That leaves you with: ((30(29))/2) = 29(15) = 435.

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