Explanation:
the 8 that study none have no impact on the other sub-groups, we only need to add them at the end for the full number of students in the group.
30 study math, 14 study math only.
that leaves 30-14 = 16 that study math and something else.
2 study math and physics only, that leaves 16-2 = 14 to study math and chemistry or all 3.
24 study physics. 8 study physics only.
that leaves 24-8 = 16 that study physics and something else.
2 study math and physics only, that leaves 16-2 = 14 to study physics and chemistry or all 3.
so, we know from the perspective of math :
14 math only
2 math and physics only
6 math and chemistry
the total for math must be 30, but we have no Venn segment missing for math.
but we are still missing 8 students of math we cannot place anywhere.
therefore, there must be a mistake in the problem description.
I assume it should be "6 study math and chemistry only".
then we can place the remaining 8 students in the central part that study all 3.
then from the perspective of physics :
8 physics only.
2 math and physics only
8 study all 3.
since we need a total of 24, there are 6 remaining, and one open Venn segment : 6 study physics and chemistry only.
from the perspective of chemistry :
6 study math and chemistry only.
6 study physics and chemistry only.
8 study all 3.
in total we need 22 students.
that are 2 remaining and one item Venn segment : 2 study chemistry only.
the total number of students is therefore :
# study math only = 14
+
# study physics only = 8
+
# study chemistry only = 2
+
# study math and physics only = 2
+
# study math and chemistry only = 6
+
# study physics and chemistry only = 6
+
# study all 3 = 8
+
# study none = 8
=
54 students in the group.
2 study chemistry only.
8 study all 3 subjects.