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Probability of energy flowing into a subsystem

Suppose we have two blocks composed of 6 harmonic oscillators each and have 3 quanta in each initially. After bringing them together, what is the probability of having all the energy move to one of the blocks?

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Let's denote the quantum states of oscillator i as n_i. So, i ranges from 1 to 12. n_i can be interpreted as the number of quanta in oscillator nr. i.

The energy of the harmonic oscillators are (n+1/2)h-bar omega. One quantum has an energy of h-bar omega. In this problem we have six quanta distributed among 12 oscillators. All possible microstates are equally likely. This means that all the possible ways we can distribute six quanta among 12 oscillators have equal probability. The probability that all the quanta are in one block is thus given by A/B, where A is the number of ways you can distribute 6 quanta to six oscillators times two, because you have two choices for which block the quanta can move to, and B is the total number of ways you can distribute the six quanta among twelve oscillators.

To calculate A and B we need to be able to count the ...

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