Problem 7 · 2018 Math Kangaroo
Easy
Counting & Probability
careful-counting
In a game of luck, a ball rolls downwards towards hammered nails and is diverted either to the right or the left by the nail immediately below it. One possible path is shown in the diagram. How many different ways are there for the ball to reach the second compartment from the left?
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Answer: C — 4
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Hint 1 of 2
Count how many rows of nails the ball passes before landing.
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Hint 2 of 2
The number of paths into each slot follows the same pattern as the rows of Pascal's triangle (1, then 1–1, then 1–2–1, …).
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Approach: count left/right choices like Pascal's triangle
- The ball passes 4 rows of nails, choosing left or right each time, and lands in one of 5 compartments.
- The number of ways to reach the compartments from left to right is 1, 4, 6, 4, 1.
- The second compartment from the left has 4 different paths.
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