Question 1126099
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<pre>
We are given 3 sets: D (dancing) of 52 elements;  B (debating) of 58 elements and W (swimming) of 64 elements.


We are also given the number of elements in each in-pair intersections.


We also are told that the union of all three sets contains 120 elements.


In such situation the following formula works


    n(D U B U W) = n(D) + n(B) + n(W) - n(D n B) - n(D n W) - n(B n W) + n(D n B n W).     (*)


Substituting the given data, you get


    120 = 52 + 58 + 64 - 14 - 27 - 32 + n(D n B n W).


In this equation, the only term  n(D n B n W)  is unknown - exactly that under the question.


So, it is easy to get it from the last equation:


    n(D n B n W) = 120 - 52 - 58 - 64 + 14 + 27 + 32 = 19.


<U>Answer</U>.  19 members enjoy all three activities.
</pre>

Solved.


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


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


    But when I added them, I counted twice every term in each in-pair intersection.


    Therefore, I subtracted the number of terms in each in-pair intersection.


    Next, when I added three first addends, I counted thrice each term in the triple intersection;

    and when I subtracted in-pair intersections, I canceled 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;(*) &nbsp;is proved &nbsp;&nbsp;// &nbsp;&nbsp;and the solution is completed &nbsp;(!&nbsp;!)


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See the lessons

&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>

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

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