A mathematics competition uses the following scoring procedure to discourage students from guessing (choosing an answer randomly) on the multiple-choice questions. For each correct response, the score is 7. For each question left unanswered, the score is 2. For each incorrect response, the score is 0. If there are 5 choices for each question, what is the minimum number of choices that the student must eliminate before it is advan- tageous to guess among the rest?

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If no answer choices are eliminated, the expected value for guessing is:

$$ 7\cdot \frac{1}{5} + 0\cdot \frac{4}{5} = \frac{7}{5} $$

This is less than the value for leaving it unanswered, so we need to eliminate something. Suppose we were able to eliminate one answer and then guess:

$$ 7\cdot \frac{1}{4} + 0\cdot \frac{3}{4} = \frac{7}{4}$$

Eliminating one more choice:

$$ 7\cdot \frac{1}{3} + 0 \cdot \frac{2}{3} = \frac{7}{3} $$

The minimum number of choices we would need to eliminate is 2.