Q:

Least Exactly Problem

I read following problem and solution:

In a recent test of $100$ students, 

$95$ answered question #1 correctly
$75$ answered question #2 correctly
$97$ answered question #3 correctly
$95$ answered question #4 correctly
$96$ answered question #5 correctly

What is the smallest number of students who could have answered exactly $4$ of the $5$ questions correctly?

The solution is:
A maximum of $75$ could have answered all questions correctly. Of the remaining $25$, the $5$ who got question #1 wrong, the $3$ who got the question #3 wrong, the $5$ who got question #4 wrong and the $4$ who got question #5 wrong could have been uniquely different giving $17$ who got exactly $3$ questions correct. This leaves $8$ who got exactly $4$ questions correct. 
My question:

What maths topic/book is this problem involved?
Is "uniquely different" a term? (it looks like $5 + 3 + 5 + 4 = 17$, but what does it mean? Or why it means who got exactly $3$ questions correct?)
Why the "least exactly $4$" problem is $8$ ($25 - 17 = 8$)?
As an extension, What is the smallest number of students who could have answered exactly $1/5, 2/5, 5/5$ questions correctly?

Thank you in advance. M

A:

I think I understood this problem now: place 5 sticks.
given 5 sticks, whose length is 25, 5, 3, 5, 4.
Here I will list 3 "extreme" cases, and the answer is obvious.
case 1: If you put all the sticks on the ground one after another, then the length is 25+5+3+5+4 = 42, which is the LARGEST number of students who got only 1 wrong. (In other words, all the other 100-42 = 58 students get all questions correct)
case 2: If you put all the sticks align at one side, then it means: 3 students get all 5 questions wrong; 4 students get 4 questions wrong; 3 students get 3 questions wrong; 20 students get 1 questions wrong; 75 students get all questions correct.
case 3: If you put stick 5, 3, 5, 4 one after another and align the head with stick 25, then: 5 + 3 + 4 + 5 = 17 students get 2 questions wrong; 25 - 17 = 8 get 1 questions wrong. 
The answer is the case #3.
Unique different means no intersection :)

