Problem:
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
Answer Choices:
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E.
Solution:
Let represent traveling a block east and represent traveling a block north. To avoid the dangerous intersection the first two blocks must be or . So there possible paths: , and .
In the following diagram, the numbers indicate the number of ways to get to each of the intersections. In each case, the number of ways to get to any particular intersection is the sum of the numbers of ways to get to any of the intersections leading directly to it. Thus, there are four paths to Jill's house, avoiding the dangerous intersection.
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Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions