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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:

  1. What maths topic/book is this problem involved?
  2. 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?)
  3. Why the "least exactly $4$" problem is $8$ ($25 - 17 = 8$)?
  4. 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

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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 :)

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  • $\begingroup$ Since no other people post answers this question, I will accept my own answer. If in the future there are better ones, I will change the accepted answer. Thanks. $\endgroup$ – milesma Dec 5 '18 at 22:31
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How I see it: the number of students getting exactly 4 questions right $(E_4)$ is embedded in the number of students getting at least 3 questions right $(A_3)$. Let $E_i$ be the number of students getting exactly i questions right. $E_4 = A_3-E_3-E_5$ and the maximal values of $A_3, E_5$ are 100 and 75 respectively and the corresponding maximal value of $E_3$ would be 17. Hence, the smallest value $A_4$ can take is 8.

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