Problem 582

For each of the 10 questions, each correct answer is worth 5 points, each unanswered question is worth 1 point, and each incorrect answer is worth 0 points.

If 10 of 10 questions are answered correctly, the total score is $10 \times 5 = 50$ points.

If 9 of 10 questions are answered correctly, either 0 or 1 questions can be unanswered. This means that the total score is either $9 \times 5 = 45$ points or $9 \times 5 + 1 = 46$ points.

If 8 of 10 questions are answered correctly, either 0 or 1 or 2 questions can be unanswered. This means that the total score is either $8 \times 5 = 40$ points or $8 \times 5 + 1 = 41$ points or $8 \times 5 + 2 = 42$ points.

If 7 of 10 questions are answered correctly, either 0 or 1 or 2 or 3 questions can be unanswered. This means that the total score is one of 35, 36, 37, or 38 points.

If 6 of 10 questions are answered correctly, either 0 or 1 or 2 or 3 or 4 questions can be unanswered. This means that the total score is one of 30, 31, 32, 33, or 34 points.

So far, we have seen that the following point totals between 30 and 50, inclusive, are possible: $$30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 45, 46, 50$$ which means that $$39, 43, 44, 47, 48, 49$$ are not possible.

If 5 or fewer questions are answered correctly, is it possible to obtain a total of at least 39 points?

The answer is no, because in this case, the number of correct answers is at most 5 and the number of unanswered questions is at most 10 (these both can’t happen at the same time) which together would give at most $5 \times 5 + 10 = 35$ points.

Therefore, there are exactly 6 integers between 30 and 50, inclusive, that are not possible total scores.