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Omni-MATH / Two players play a game where they are each given 10 indistinguishable units…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

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 the players can be expressed as ab\frac{a}{b}, where a,ba, b are relatively prime positive integers. Compute 100a+b100a+b.
Plain-text mathematical notation (without MathML)
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 the players can be expressed as (a)/(b), where a,b are relatively prime positive integers. Compute 100a+b.
Original LaTeX notation
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 the players can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100a+b$.

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Source and history

task_id
bb15bb85-3e72-55dc-bdaa-8f4be13d9a3d
task_key
test--bb15bb85-3e72-55dc-bdaa-8f4be13d9a3d
task_revision_id
1
config / split
/ test

Official source

initial import