SOLUTION: There are 25 students asked by their literature instructor regarding with the type of literary works they prefer to read. He found out that 10 prefer to read novels, 11 prefer t

Use the "inclusion-exclusion principle".


It is the formula, which allows to find the number of elements in the union of three subsets

of the given universal set.


In your case, you are given the universal set of 25 students, and three its "basic" subsets of 10, 11 and 15 students.


You also are given their in-pair intersections of 5, 4 and 7 students.


Finally, you are given their triple intersection of 3 students.


The formula of "inclusion-exclusion principle" says:


    +-----------------------------------------------------------------+
    |      TO FIND THE NUMBER OF ELEMENTS in the UNION                |
    |                                                                 |
    |           of the three basic subsets                            |
    |                                                                 |
    |     add the number of elements in the basic subsets,            |
    |     subtract the numbers of elements of in-pair intersections   |
    |     and add the number of elements in triple intersection       |
    +-----------------------------------------------------------------+


According to this rule (the formula), the union contains  

    10 + 11 + 15 - 5 - 4 - 7 + 3 = 23 students.


The rest are  25 - 23 = 2 students that prefer none of the given types of literary works.     ANSWER

    First, it should be totally clear to you why I add the first three basic addends.

    But when I add them, I count twice the terms in each in-pair intersection.

    Therefore, I subtract the numbers of terms in each in-pair intersection.

        Next, when I add three basic addends, I count thrice each term in the triple intersection;

        and when I subtract in-pair intersections, I cancel these terms thrice.

    Therefore, I must add the number of terms in the triple intersection one more time to restore the balance.