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HMMT 2010 C6

  • Combinatorics
  • Answer

An ant starts out at (0,0)(0,0). Each second, if it is currently at the square (x,y)(x, y), it can move to (x−1,y−1),(x−1,y+1),(x+1,y−1)(x-1, y-1),(x-1, y+1),(x+1, y-1), or (x+1,y+1)(x+1, y+1). In how many ways can it end up at (2010,2010)(2010,2010) after 4020 seconds?

Dataset text NuminaMath olympiads. Quoted with source.

Also HMMT 2010 F8.

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\binom{4020}{1005}^2

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