Question 1095656
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<pre>
Let  A  be the subset of those 10 who would not go to a&#8203; park;

let  B  be the subset of those 17 who would not go to a&#8203; beach;  and

let  C  be the subset of those 12 who would not go to a&#8203; cottage.


Then we are given that 

    - the intersection AB of the subsets A and B  consists of 3 persons (="neither a park nor a&#8203; beach");

    - the intersection BC of the subsets B and C  consists of 11 persons (="neither a beach nor a&#8203; cottage");

    - the intersection AC of the subsets A and C  consists of 2 persons (="neither a park nor a&#8203; cottage").


We are also given that the intersection ABC of the subsets A, B and C consists of 1 person ("would not go to a park or a beach or a&#8203; cottage"),   and

the supplement of the UNION of the sets A, B and C to the entire group consists of 2 persons ("willing to go to all three places").



Now, there is a FUNDAMENTAL and ELEMENTARY formula in the theory of finite sets saying that

    n(A U B U C) = nA + nB + nC - nAB - nBC - nAC + nABC.

Here the small letter n before the set/the subset name means "the number of elements in the subset", 

    or, using the high level terminology, "the cardinality" of the finite subset.



I will not distract your attention now for proving this formula.
Instead, I will show you how to solve the problem in two lines, using this formula.


    Line 1:  n(A U B U C) = 10 + 17 + 12 - 3 - 11 - 2 + 1 = 24 persons in the UNION  (A U B U C),   and

    Line 2:  The entire set = (A U B U C) + 2 persons   has  24 + 2 = 26 persons in total.


<U>Answer</U>.  The entire group consists of 26 persons.
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Instead of proving the formula &nbsp;(*), &nbsp;I'll direct you to my 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>


You will find there all the arguments needed for the proof &nbsp;(and, actually, the proof itself) &nbsp;in the entertainment form.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Read it and have fun !