In a counting transition from AMC 12 to AIME practice, a useful goal is learning to build the restriction into the representation. Counting everything and subtracting a few obvious bad cases may leave hidden overlaps.
The original exercise below is a foundation example, not a released AIME question or an official difficulty benchmark.
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Define the objects and the restriction
Count binary strings of length eight containing exactly three ones, with no two ones adjacent. Positions are labelled from left to right, so rotations are not identified.
Without the adjacency condition, there would be C(8,3) = 56 strings. We need a representation that keeps each pair of ones separated.
Method 1: compress the positions
Let the positions of the ones be p1 < p2 < p3, with p2 at least p1 + 2 and p3 at least p2 + 2. Set q1 = p1, q2 = p2 - 1 and q3 = p3 - 2.
Then 1 ≤ q1 < q2 < q3 ≤ 6. Choosing any three distinct positions from 1 through 6 determines one permitted original string by reversing the shifts. The count is C(6,3) = 20.
Method 2: distribute the remaining zeros
There are five zeros. Reserve one zero in each of the two internal gaps between the three ones. Three zeros remain to distribute among the four gaps: before, between, between and after.
The number of nonnegative gap allocations is C(6,3) = 20. Explain why each allocation creates exactly one string, rather than treating the formula as a substitute for the correspondence.
Test a circular variation
If the first and last positions are also considered adjacent, the original count includes four forbidden strings with ones at both endpoints. The third one can then occupy position 3, 4, 5 or 6.
Removing those gives 16 for this labelled-position circular-adjacency variation. This is not yet a count up to rotation; identifying rotated strings would be a separate problem.
Make the bridge skill explicit
The transferable lesson is to choose a representation in which constraints are automatic, and then prove the mapping is reversible. That prevents both omissions and duplicate counting.
After a delayed redo, change the length or number of ones and derive the new bounds. For a discussion of counting difficulties, send AMC12 AIME COUNTING with an attempt showing how you handled the restriction. Official competition rules should be checked separately.
Continue reading: From AMC 12 Geometry to AIME Practice: Turn a Sketch into an Exact Model.
Contact Hanlin
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