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Each student studies at least 1 language. If it is also known that exa 04 Oct 2019, 03:15
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78% (01:29) correct 22% (01:52) wrong based on 231 sessions

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In a study group of six students, each student is studying at least one of the following languages: Russian, Ukrainian, or Hebrew. The distribution of language studies is as follows:

• Russian is studied by four students.

• Ukrainian is studied by three students.

• Hebrew is studied by two students.

If three students from the group are studying exactly two languages, how many students study all three languages?


A. 0
B. 1
C. 2
D. 3
E. 4

M04-19

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Each student studies at least 1 language. If it is also known that exa 04 Oct 2019, 03:24
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6=4+3+2-3-all three
all three; 0
IMO A

Bunuel
6 students in a group study different languages as specified:

Russian: 4

Ukrainian: 3

Hebrew: 2

Each student studies at least 1 language. If it is also known that exactly 3 students study exactly 2 languages, how many students are studying all three languages?


A. 0
B. 1
C. 2
D. 3
E. 4

M04-19
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Each student studies at least 1 language. If it is also known that exa 01 Sep 2020, 15:36
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I did this:

Total number of students = Students speaking only 1 language - 2* Students speaking exactly 2 languages - Students speaking only 3 languages.

6 = 3 - 2*3 - all
0 = -all
all = 0.
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Each student studies at least 1 language. If it is also known that exa 12 May 2023, 12:40
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In a study group of six students, each student is studying at least one of the following languages: Russian, Ukrainian, or Hebrew. The distribution of language studies is as follows:

• Russian is studied by four students.

• Ukrainian is studied by three students.

• Hebrew is studied by two students.

If three students from the group are studying exactly two languages, how many students study all three languages?


A. 0
B. 1
C. 2
D. 3
E. 4


The formula for calculating the number of students in three overlapping sets is as follows:

(Total number of students) = (Students studying Russian) + (Students studying Ukrainian) + (Students studying Hebrew) - (Students studying exactly two languages) - 2*(Students studying all three languages) + (Students studying none of the languages).

When we initially add up the students studying Russian, Ukrainian, and Hebrew, the counts have overlaps. For instance, students studying exactly two languages end up being counted twice since they fall into two language groups. Similarly, students studying all three languages get counted three times, once per language group. To correct this, we adjust the total by subtracting the count of students studying exactly two languages once and the count of students studying all three languages twice. This adjustment ensures that each student, regardless of the number of languages they're studying, is only counted once.

For more information and examples, check out this link: ADVANCED OVERLAPPING SETS PROBLEMS

Applying the formula to our case, we have: \(6=4+3+2-3-2*x+0\);

Solving gives \(x = 0\).

Thus, we find that no students are studying all three languages.


Answer: A
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