A multiple choice test has 10 questions on it. Each question answered correctly is worth 5 points, each unanswered question is worth 1 point, and each question answered incorrectly is worth 0 points. How many of the integers between 30 and 50 , inclusive, are not possible total scores?
(A) 2
(B) 3
(C) 4
(D) 6
(E) 5
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.