SOLUTION: a survey of 65 people was conducted during a seafood night in a local community.25 people like lobster,30 like fish and 19 like whelks.4 liked lobster and whelks only,2 liked whelk

In this problem, we have an "universal" set of 65 people and several its subsets.


Subset L of 25 people who like lobster;

Subset F of 30 people who like fish;

Subset W of 19 people who like whelks;


Subset LWo of 4 persons who like lobster and whelks only;

Subset WFo of 2 persons who like whelks and fish only;

Subset FLo of 3 persons who like fish and lobster and only;


Subset N of 6 persons who did not like any of the 3 choices.



Let LFW be the subset of people who like all three choices, and let x be the number of people in this subset.


Then (LWo U LWF) is the disjoint union, equal to the intersection LW with (4+x) members;

     (WFo U LWF) is the disjoint union, equal to the intersection WF with (2+x) members;

     (FLo U LWF) is the disjoint union, equal to the intersection FL with (3+x) members.


Now apply the Inclusive-Exclusive principle, which says

    |(L U F U W)| = |L| + |F| + |W| - |LW| - |WF| - |FL| + |LFW|


Substitute here

    |(L U F U W)| = 65 - 6 = 59;

    |L| = 25,  |F| = 30,  |W| = 19,  |LW| = 4+x,  |WF| = 2+x,  |FL| = 3+x,  |LFW| = x.


Equation (1) becomes

    59 = 25 + 30 + 19 - (4+x) - (2+x) - (3+x) + x,

    59 = 65 - 2x,

    2x = 65 - 59 = 6,

     x           = 6/2 = 3.


ANSWER.  The number of those who like all 3 choices is 3.