Math Gold Medalist

Lor

2023 AIME I

Problem 11

Find the number of subsets of $\{1,2,3,\ldots,10\}$ that contain exactly one pair of consecutive integers. Examples of such subsets are $\{\mathbf{1},\mathbf{2},5\}$ and $\{1,3,\mathbf{6},\mathbf{7},10\}.$

Recursive Sequence

Casework

an: In how many ways can we select some non-adjacent elements from 1 to n

Find recursive formula of an

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   Solution