SOLUTION: A survey of 539 adults aged 18-24 year olds was conducted in which they were asked what they did last Friday night. It found: 173 watched TV 176 hung out with friends 27 watc

We are given a universal set U of 539 persons and its subsets

T  (watched TV)                                                          of 173

F  (hung out with friends)                                               of 176

TP \ TPF  (watched TV and ate pizza, but did not hang out with friends)  of  27

TF \ TPF  (watched TV and hung out with friends, but did not eat pizza)  of  41

FP \ TPF  (hung out with friends and ate pizza,  but did not watch TV)   of  26 

TPF       (watched TV, hung out with friends, and ate pizza)             of  39

The complement of (T U F U P) to U                                       of  73.


Couple comments regarding this table.

    For brevity, I write AB for the intersection of subsets A and B (instead of traditional (A ∩ B) ).
    For brevity, I write TPF for the triple intersection (T ∩ P ∩ F).
    The sign  " \ "  means subtraction of subsets.



Now, in the elementary set theory, there is a REMARCABLE identity 

    n(A U B U C) = n(A) + n(B) + n(C) - n(AB) - n(AC) - nBC) + n(ABC)              (1)

for any three subsets A, B, C of a universal set, their in-pair intersections AB, AC and BC, and the triple intersection ABC.



I will apply it for the given subsets T, F, P, their in-pairs and triple intersetions

    n(T U F U P) = n(T) + n(F) + n(P) - n(TF)   - n(TP)   - n(FP)   + n(TPF).      (2)



In this equality, all the terms are given and are known, EXCEPT only one n(P).  It is my goal now to find n(P).
For it, I substitute all the known values into equation (2)

    539 - 73     = 173  + 176  + n(P) - (41+39) - (27+39) - (26+39) + 39.          (3)



From (3) I get

    n(P) = (539-73) - 173 - 176 + (41+39) + (27+39) + (26+39) - 39 = one click in my Excel = 289.


Now my last step is

    the number of those who only eat Pizza = n(P) - n(TP) - n(FP) + n(TPF) = 289 - (27+39) - (26+39) + 39 = 197.


ANSWER.  The number of those who only eat Pizza is 197.