Particle Interaction

Probability Level 4

Suppose we have 2016 particles in space. 671 of these particles are of type A, 672 are of type B, and 673 are of type C. Any time two particles of different types interact, both of them change into the third type. After some time, which of these ordered triples can denote the number of particles of each type?

Notes:

Assume that, given enough time, the particles will interact.

Each answer is of the form ( A , B , C ) (A, B, C) . In this ordered triple, A A denotes the number of particles of type A, B B denotes the number of particles of type B, and C C denotes the number of particles of type C.

( 494 , 806 , 716 ) (494, 806, 716) ( 1 , 1005 , 1010 ) (1, 1005, 1010) None of these choices ( 541 , 1267 , 208 ) (541, 1267, 208) At least two of these choices ( 2016 , 0 , 0 ) (2016, 0, 0) ( 950 , 721 , 345 ) (950, 721, 345) ( 672 , 671 , 673 ) (672, 671, 673)

Steven Yuan by Steven Yuan , 20, USA

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1 solution

Notice that whenever two particles interact, each of a-b, b-c, and c-a mod 3 must stay the same, as subtracting 1 from a number mod 3 is the same as adding 2 to a number mod 3. None of the answer choices obey this, so we can conclude that the correct answer is none of the above.

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