PUMaC 2015 · 组合（B 组） · 第 4 题

PUMaC 2015 — Combinatorics (Division B) — Problem 4

[ 4 ] Andrew has 10 balls in a bag, each a different color. He randomly picks a ball from the bag 4 times, with replacement. The expected number of distinct colors among the balls he picks p is , where gcd( p, q ) = 1 and p, q > 0. What is p + q ? q

解析

Now, if a black does not have a diagonal of black cubes (allowing wrap-arounds), it must contain at least 4 cubes, so there are at least two blocks with diagonals and with exactly 3 cubes. We consider two cases. Case 1: The diagonals of these two blocks are oriented in the same direction. Clearly, the third block must contain a diagonal oriented in the same direction as well. The remaining black cube can be anywhere else in the block. There are 3 · 6 · 2 = 36 ways to choose the first two blocks and their diagonals. There are 1 · 6 = 6 ways to choose black cubes in the remaining block. This gives a total of 216 colorings. Case 2: They are oriented in opposite directions. Then, the black cubes in the remaining block is determined (consider the projection of the blocks on top of one another; four squares are missing and the remaining block contains four black cubes). There are 3 · 6 · 3 = 54 ways to choose the first to blocks and their diagonals. There is only 1 way to choose the black cubes in the remaining block. This gives a total of 54 colorings. In total, then, there are 216 + 54 = 270 ways to choose 10 of the smaller cubes to paint black. Author: Bill Huang