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Suppose a multiple choice question has $n$ answers. You allow a student $m$ tries. The student gets the correct answer on the $k$-th try. What is a "fair" score?

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    $\begingroup$ This is subjective and different scales will be best for different purposes. One interesting theoretical approach might be to consider the information content implied by their answer. (Look up entropy and information theory.) $\endgroup$ – aes Apr 17 '15 at 15:33
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More context is needed to establish what is "fair". To illustrate this, take an extreme example. Say the question is "What is $1+1$?", and $99$ ridiculous answers are provided, along with the correct one, and $50$ tries are allowed. One student needs $10$ attempts to get it right, while another needs $40$. It's hard to argue that the first student has demonstrated more knowledge than the second one. In this case, the only reasonable scoring system would be $1$ point (or whatever) for a correct first answer and no credit otherwise. The situation would be different if the question were hard and the answers were plausible but required graded levels of knowledge to eliminate (or confirm) them. For these reasons, I don't think that the posted question, as it stands, can be given a mathematical answer.

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You could have a scoring system where the problem is worth $n$ points. You can guess as many options (choices) as you want. You get $1$ point for each correctly chosen correct option and $1$ point for each nonchoice of each incorrect option.

Thus in your scenario you would score $1 +(n-k)$ (one correct choice of the correct option; and $n-k$ correct non-choices of incorrect options).

So the score would be $n+1-k$ out of a possible $n$ points.

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