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16 Sep 2014, 00:42
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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?
(1) There are no students studying all three of the subjects.
(2) None of those who study math study chemistry.
(1) There are no students studying all three of the subjects.
(2) None of those who study math study chemistry.
16 Sep 2014, 00:42
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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
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
General Discussion
12 Oct 2015, 16:50
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sorry for not so clear image.
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