Let  A  be the subset of those 10 who would not go to a​ park;

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

let  C  be the subset of those 12 who would not go to a​ cottage.


Then we are given that 

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

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

    - the intersection AC of the subsets A and C  consists of 2 persons (="neither a park nor a​ 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​ 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.


Answer.  The entire group consists of 26 persons.