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But you can use those 12 bits as a key into some pre-arranged set of information, each of which carries more information.

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Actually you can't. There is no prior information about the distribution of correct answers given in the question so everything is uniform distribution and there is nothing to set up "pre-arranged" entropy encoding with.

The trick (to get around the information theoretic limit of 12) is to be uncertain about which of the answers are correct.

For example, with 1 bit, you can transmit if the majority are 1 or 0, already getting you at least 11 right answers but no information about which ones are right.

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