We have a universal set of 100 students and the following subsets

    M   studied Math         (42)

    P   studied Psychology   (68)

    H   studied History      (54)

    M ∩ H  studied Math and History     (22)

    M ∩ P  studied Math and Psychology  (25)

    Ho   studied History and neither Math nor Psychology  (7)   ( = H \ (H ∩ M) \ (H ∩ P) )

    M ∩ P ∩ H  studied all 3 subjects

    U \ (M U P U H)  did not take any of the three subjects  (8)


So, from the condition, the union  (M U P U H) of those who study at least one of the three subjects, has  100-8 = 92 students.


They want you determine the number of students who take History and Psychology but not Mathematics.

This set is  (H ∩ P) \ (H ∩ P ∩ M).



To solve the problem, we will find first the number of students in the set  (H ∩ P), and then will subtract 10 (which is  (H ∩ P ∩ M) )  from it.



To find the number of students in the set  (H ∩ P), use this remarkable identity from the elementary set theory

    n(M U P U H) = n(M) + n(P) + n(H) - n(M ∩ P) - n(M ∩ H) - n(P ∩ H) + n(M ∩ P ∩ H).    (1)


This identity says that the number of element in the union of any three subsets of a universal set is the sum of elements 
in each subset minus the sum of elements in in-pair intersections of the subsets plus the number of elements in the triple 
intersection of the three subsets.

This identity is valid for any three subsets of a universal set.



Substitute the given value into the identity (1).  You will get then

    92 = 42 + 68 + 54 - 22 - 25 - n(H ∩ P) + 10.


It implies 

    n(H ∩ P) = 42 + 68 + 54 - 22 - 25 + 10 - 92 = 35.


The last step to get the answer is to subtract 10 from 35:  

    the number of students who take History and Psychology but not Mathematics = n((H ∩ P) - n(M ∩ P ∩ H) = 35 - 10 = 25.    


So the probability the problem asks for is  P =  = 0.25.    ANSWER