Problem 12 · 2021 Math Kangaroo
Stretch
Counting & Probability
careful-counting
Ahmad walks up 8 steps, going up either 1 or 2 steps at a time. There is a hole on the 6th step, so he cannot use this step. In how many different ways can Ahmad reach the top step?
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Answer: C — 8
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Hint 1 of 2
Count, step by step, the number of ways to reach each step using 1- or 2-step moves.
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Hint 2 of 2
The forbidden 6th step has 0 ways; carry that 0 forward to the top.
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Approach: build up ways-to-reach each step (Fibonacci-style)
- Ways to reach steps 1..5 are 1, 2, 3, 5, 8 (each is the sum of the two below).
- Step 6 is a hole, so it has 0 ways and cannot be used.
- Step 7 = ways(5)+ways(6) = 8+0 = 8; step 8 = ways(6)+ways(7) = 0+8 = 8.
- So there are 8 ways, choice (C).
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