HMMT 2008 T3

Team 1
Answer

Let $n>4$ be a positive integer. Determine the number of ways to walk from $(0,0)$ to $(n, 2)$ using only up and right unit steps such that the path does not meet the lines $y=x$ or $y=x-n+2$ except at the start and at the end.

Dataset text NuminaMath olympiads. Quoted with source.

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\frac{1}{2}\left(n^{2}-5 n+2\right)

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