Question 1163106
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<pre>

You are given the sets 

    U - universal set of all 440 students
    E - set of students passed English (300)
    S - set of students passed Spanish (280)
    M - set of students passed Math    (220)

the in-pair intersections

    ES - English and Spanish           (180)
    SM - Spanish and Math              (160)
    EM - English and Math              (150)

and the triple intersection

    ESM - English, Spanish and Math    (100)


For the union E U S U M, there is a REMARKABLE formula from elementary set theory 


    n(E U S U M) = n(E) + n(S) + n(M) - N(ES) - n(SM) - n(EM) + n(ESM).    (1)


It says the number of elements in the union of three subsets is the alternate sum of the shown components.


By substituting the given values, you get the answer to the problem's question


    n(E U S U M) = 300 + 280 + 220 - 180 - 160 - 150 + 100 = 410.


<U>ANSWER</U>.  In all, 410 students passed at least one exam.
</pre>

Solved.


Memorize the formula and the method of the solution (!)


=================


The last step is to prove the formula (1).


<pre>
    It is totally clear to you why I add the first three addends in the formula (1).

    But when I add them, I count twice every term in each in-pair intersection.

    Therefore, I subtract the numbers of terms in each in-pair intersection.

    Next, when I add three first addends, I count thrice each term in the triple intersection;

    and when I subtract in-pair intersections, I cancel these terms thrice.

    Therefore, I must add the number of terms in the triple intersection one more time to restore the balance.
</pre>

Thus the formula &nbsp;(1) &nbsp;is proved &nbsp;&nbsp;// &nbsp;&nbsp;and the solution is fully completed &nbsp;(!&nbsp;!)



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https://www.algebra.com/algebra/homework/word/evaluation/Evaluation_Word_Problems.faq.question.1126099.html


to similar solved problems in the archive of this forum.




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To tutor @greenestamps:


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Thanks for noticing my typos ---  I just fixed them (!)