HMMT 十一月 2020 · 冲刺赛 · 第 26 题

HMMT November 2020 — Guts Round — Problem 26

[13] Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins at that location if the number of units they placed there is at least 2 more than the units of the other player. If both players distribute their units randomly (i.e. there is an equal probability of them distributing their units for any attainable distribution across the 3 locations), the probability that at least one location is won by one of a the players can be expressed as , where a, b are relatively prime positive integers. Compute 100 a + b . b

解析

[13] Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins at that location if the number of units they placed there is at least 2 more than the units of the other player. If both players distribute their units randomly (i.e. there is an equal probability of them distributing their units for any attainable distribution across the 3 locations), the probability that at least one a location is won by one of the players can be expressed as , where a, b are relatively prime positive b integers. Compute 100 a + b . Proposed by: Sheldon Kieren Tan Answer: 1011 ( ) 2 12 2 Solution: By stars and bars, the total number of distributions is = 66 . If no locations are won, 2 either both distributions are identical or the difference between the two is (1 , 0 , − 1), in some order. The first case has 66 possibilities. If the difference is (1 , 0 , − 1), we can construct all such possiblities by choosing nonnegative integers a, b, c that sum to 9, and having the two players choose ( a + 1 , b, c ) ( ) 11 and ( a, b, c + 1). This can be done in = 55 ways. In total, the second case has 6 · 55 = 5 · 66 2 possibilities. 6 · 66 1 10 Thus the probability that no locations are won is = , meaning that the answer is . 2 66 11 11