# Peculiar nature of the different marks obtainable in a test

combinationsnumber theory

In an examination of $$20$$ questions, a student gets $$-1,0,4$$ marks for incorrect, unattempted, and correct answers respectively. Let $$S$$ be set of all the marks that a student can score. What is the number of distinct elements in $$S$$?

In the range of $$101$$ marks $$-20,-19,\cdots80$$, the marks $$\{+79,+78,+77,+74,+73,+69\}$$ are unobtainable, leaving us $$95$$ elements in $$S$$.

I find this rather strange. What is so special about this set of marks that makes them unobtainable? I couldn't find any pattern in them, but they are very close to each other, lying around $$70$$.

From a perfect score of $$80$$, for every unmarked or wrong question, $$4$$ or $$5$$ points are lost.

$$20$$ correct $$\to$$ score $$\in \{80\}$$.

$$19$$ correct $$\to$$ score $$\in \{75,76\}$$

$$18$$ correct $$\to$$ score $$\in \{70,71,72\}$$

$$17$$ correct $$\to$$ score $$\in \{65,66,67,68\}$$

$$16$$ correct $$\to$$ score $$\in\{60,61,62,63,64\}$$

And from now on all lower numbers will be achieved.