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# M10-17

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Math Expert
Joined: 02 Sep 2009
Posts: 59687

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16 Sep 2014, 00:42
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Difficulty:

25% (medium)

Question Stats:

77% (01:49) correct 23% (01:41) wrong based on 71 sessions

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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 of the three subjects.

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

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16 Sep 2014, 00:42
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Official Solution:

Statements (1) and (2) combined are not sufficient. Consider the situation when those who study math do not study any of the other two subjects and when
1. all those who study chemistry also study physics ($$20 - 5 - 10 = 5$$, there are 5 students who do not study any of the three subjects);
2. only one student studies both chemistry and physics ($$20 - 5 - 15 = 0$$, there are no students who do not study any of the three subjects).

Alternative Explanation:

First of all note that total # 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} &gt; 0? Or: is {people in none of the groups} = {people in exactly 2 groups} + 2*{people in exactly 3 groups} - 1 &gt; 0

(1) There are no students studying all of the three 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 &gt; 0. Now, if {people in exactly 2 groups} = 1 then the answer will be NO (for example if there is only one student who study 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.

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12 Oct 2015, 16:50
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sorry for not so clear image.
Attachment:
New Doc 3_2.jpg

Attachment:
New Doc 3_1.jpg

>> !!!

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M10-17   [#permalink] 12 Oct 2015, 16:50
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# M10-17

Moderators: chetan2u, Bunuel