M10-17

Official Solution:


If among 20 students in a group, 5 study math, 10 study physics, and 6 study chemistry, are there any students who do not study any of the above-mentioned subjects?

First, note that the total number of students is 20. Next:

Total = {people in group A} + {people in group B} + {people in group C} - {people in exactly 2 groups} - 2*{people in exactly 3 groups} + {people in none of the groups}.

20 = 5 + 10 + 6 - {people in exactly 2 groups} - 2*{people in exactly 3 groups} + {people in none of the groups}.

So: {people in none of the groups} = {people in exactly 2 groups} + 2*{people in exactly 3 groups} - 1.

Question: is {people in none of the groups} > 0?

Or: is {people in exactly 2 groups} + 2*{people in exactly 3 groups} - 1 > 0

(1) There are no students studying all three of the subjects.

So, {people in exactly 3 groups} = 0. Hence, the question becomes: is {people in none of the groups} = {people in exactly 2 groups} - 1 > 0? Now, if {people in exactly 2 groups} = 1, then the answer will be NO (for example, if there is only one student who studies exactly two subjects: math and physics, and all other students study only one subject). But if {people in exactly 2 groups} = 2, then the answer will be YES (for example, if there are two students who study exactly two subjects: math and physics, and all other students study only one subject). Not sufficient.

(2) None of those who study math study chemistry. Clearly insufficient.

(1)+(2) Examples from (1) are still valid; thus, we have both YES and NO answers. Not sufficient.


Answer: E