PUMaC 2008 · 组合（A 组） · 第 8 题

PUMaC 2008 — Combinatorics (Division A) — Problem 8

(5 points) SET cards have four characteristics: number, color, shape, and shading, each of which has 3 values. A SET deck has 81 cards, one for each combination of these values. A SET is three cards such that, for each characteristic, the values of the three cards for that characteristics are either all the same or all different. In how many ways can you replace each SET card in the deck with another SET card (possibly the same), with no card used twice, such that any three cards that were a SET before are still a SET? (Alternately, a SET card is an ordered 4-tuple of 0s, 1s, and 2s, and three cards form a SET if their sum is (0 , 0 , 0 , 0) mod 3; for instance, (0 , 1 , 2 , 2), (1 , 0 , 2 , 1), and (2 , 2 , 2 , 0) form a SET. How many permutations of the SET cards maintain SET-ness?) 9. (7 points) How many spanning trees does the following graph (with 6 vertices and 9 edges) have? (A spanning tree is a subset of edges that spans all of the vertices of the original graph, but does not contain any cycles.) E D F C A B

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