PUMaC 2015 · 组合（B 组） · 第 7 题

PUMaC 2015 — Combinatorics (Division B) — Problem 7

[ 7 ] Alice has an orange 3-by-3-by-3 cube, which is comprised of 27 distinguishable, 1-by-1-by-1 cubes. Each small cube was initially orange, but Alice painted 10 of the small cubes completely black. In how many ways could she have chosen 10 of these smaller cubes to paint black such that every one of the 27 3-by-1-by-1 sub-blocks of the 3-by-3-by-3 cube contains at least one small black cube?

解析

[ 7 ] Alice has an orange 3-by-3-by-3 cube, which is comprised of 27 distinguishable, 1-by-1-by-1 cubes. Each small cube was initially orange, but Alice painted 10 of the small cubes completely black. In how many ways could she have chosen 10 of these smaller cubes to paint black such that every one of the 27 3-by-1-by-1 sub-blocks of the 3-by-3-by-3 cube contains at least one small black cube? Solution: Divide the 3-by-3-by-3 cube into 3 1-by-3-by-3 blocks. If 10 total smaller cubes are painted black, then two of these blocks must contain 3 black cubes and the third contains