Problem 14 · 2018 Math Kangaroo
Hard
Counting & Probability
careful-counting
On one day there are 40 train trips, each from one of the towns M, N, O, P, Q to exactly one other of those towns. There are 10 trips either from or to M, 10 either from or to N, 10 either from or to O, and 10 either from or to P. How many trips are there either from or to Q?
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Answer: E — 40
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Hint 1 of 2
Each trip touches exactly two towns, so summing “trips touching each town” counts every trip twice.
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Hint 2 of 2
The five town-counts must add up to 2 × 40 = 80.
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Approach: each trip is counted at both of its towns
- Every trip has two endpoints, so adding the counts for all five towns gives \(2\times 40 = 80\).
- M, N, O and P account for \(10+10+10+10 = 40\).
- So Q must account for \(80-40 = \) 40.
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