A or B or C = A + B + C - AB - AC - BC + ABC
in your problem, this becomes:
A or B or C = 10 + 8 + 6 - 3 - 4 - 0 + 1 = 18.
the problem states 20 students, 16 of which answered at least 1 of the problem correctly.
these numbers indicate that 18 students answered 1 of the problems correctly.
it does appear that you are missing a category, namely BC.
i put a 2 in the BC category and then the numbers came out correctly.
A or B or C = A + B + C - AB - AC - BC + ABC
with 2 in BC category, formula becomes:
A or B or C = 10 + 8 + 6 - 3 - 4 - 2 + 1 = 16
add the 4 that didn't answer any correctly and you get a total of 20.
bottom line is your problem doesn't come out correctly because there is something wrong with the numbers used.
i used the following venn diagram generator to confirm that the number don't come out correct as shown, but do come out correct when i put a 2 in the BC category.
first 2 displays are with numbers as given.
next 2 displays are with numbers as given plus BC category has 2 in it.
to show you how it works logically, i'll use the numbers given, but will add a category of BC = 2.
you are given, plus what i added:
total students = 20
total students who got at least 1 of the problems correct = 16
total who got none correct = 4 (i created this category).
A = 10
B = 8
C = 6
AB = 3
AC = 4
BC = 2 (this is what i added to make the problem come out corect).
ABC = 1
ABC is pure since there are no other categories included in it.
from AB and AC and BC, subtract ABC pure to get:
AB pure = 2
AC pure = 3
BC pure = 1
ABC pure = 1
From A, subtract AB pure, AC pure, ABC pure, to get A pure = 10-2-3-1 = 4
from B, subtract AB pure, BC pure, ABC pure, to get B pure = 8-2-1-1 = 4
from C, subtract AC pure, BC pure, ABC pure, to get C pure = 6-3-1-1 = 1
your pure categories are:
none pure = 4
A pure = 4
B pure = 4
C pure = 1
AB pure = 2
AC pure = 3
BC pure = 1
ABC pure = 1
add them up and you get a total of 20, as you should.
you can do it by formula or you can do it by logic.
the total should be 20 students.
it is, but only after i added the category of BC with 2 in it.
bottom line:
you won't get a good answer with the numbers as given, but you will get a good answer with the number as given plus the numbers i added into the BC category.
some reference that you might find helpful.
https://www.easycalculation.com/algebra/venn-diagram-3sets.php
https://www.thoughtco.com/probability-union-of-three-sets-more-3126263
http://www.probabilityformula.org/union-of-events.html