# 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:

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?

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 :)

• 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. – milesma Dec 5 '18 at 22:31

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.