TITLE: Number of different possible game scores on a quiz game with n questions
QUESTION [7 upvotes]: On a quiz game, players get 5 points for a correct answer, 2 points for not answering, and 0 points for a wrong answer. If there are n questions, find a way to determine how many possible scores there are.
I'm thinking this isn't as hard as I think it is. All even numbers are a multiple of 2, and odd numbers ≥5 are given by 5 + multiples of 2. Would it be all integers between the maximum possible score (5n) and 0, discard scores 1 and 3? 
Even in this way, though, don't know how the 0 points for a wrong answer would be incorporated.
Edit: 1 and 3 don't work. I think 5n -1 , 5n - 2, 5n - 4 and 5n-7 don't work for a total of 6 solutions that don't work? So 5n - 6

REPLY [2 votes]: On a quiz with $n$ questions, the score will be between 0 and $5n$ so there are at most $5n+1$ possibilities.
As observed above, scores of 1 and 3 are not possible at the ''bad'' end of the score range. If we instead consider the ''good'' end of possible high scores, we can observe the following. For any question not answered we lose 3 points, and for every incorrect answer we lose 5 points. Thus the scores
$$5n - 1,\, 5n-2,\, 5n-4,  \quad\text{and} \quad5n-7$$
are not possible, since the numbers $1, 2, 4,$ and $7$ cannot be written as a sum of $3$'s and $5$'s.
In all we have found 6 scores that are not possible: 2 in the ''bad'' range, and 4 in the ''good'' range. If we additionally assume that $5n - 7 > 3$, i.e. $n > 2$, then these are in fact 6 distinct scores that are not possible so there are at most $5n-5$ scores which are possible.
I'll make the claim that all remaining $5n-5$ scores are possible, and leave this proof as an exercise. 
Thus the answer is $5n-5$ when $n\geq 3$, and  the remaining cases can be computed separately: 6 possibilities with 2 questions and 3 possibilities with 1 question.