Question 1085902
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<pre>
Let "a" be the set of students that answered the question A correctly.

Let "b" be the set of students that answered the question B correctly.

Let "c" be the set of students that answered the question C correctly.



Let ab be the set of students that answered both question A and B correctly.

Let bc be the set of students that answered both question B and C correctly.

Let ac be the set of students that answered both question A and C correctly.



Let abc be the set of students that answered all three question A, B and C correctly.



Next, for any finite set "S" in this problem i will denote by |S| the number of its elements.



From the elementary set theory, this formula is well known

|a U b U c| = |a| + |b| + |c| - |ab| - |ac| - |ab| + |abc|    (1)    (see the reference at the end of my post)


In our case  |a U b U c| = 16: it is the set of students who correctly answered at least one question.



Further, from the condition

|a| = 10,  |b| = 8,  |c| = 6,  |ab| = 3,  |ac| = 4,  |abc| = 1.



The value |bc| is unknown, and we EASILY will find it from the equation (1) after substituting all other known values to the equation:

16 = 10 + 8 + 6 - 3 - 4 - |bc| + 1.


It gives |bc| = 10 + 8 + 6 - 3 - 4 + 1 - 16 = 2.
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Solved.


<U>Answer</U>.  2 students answered correctly questions B and C.



Regarding the formula (1), read these two lessons in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/misc/Counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Counting elements in sub-sets of a given finite set</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/Advanced-probs-counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Advanced problems on counting elements in sub-sets of a given finite set</A>



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic 
"<U>Miscellaneous word problems</U>".



Ignore everything the tutor "Theo" wrote in his post: it is not relevant . . . Unfortunately.



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Usually, I read attentively the students' posts, as well as the tutors' posts.

(Except statistic problems, where I am not an expert).


But I never read the tutor's Theo posts: they always are so long that are simply unreadable.


I do not believe that the right Math problem may have long formulation. 


I also do not believe that the right solution to elementary Math problem should be long.