## Task 1 - 190821

A group of 39 students is discussing their grades in the subjects of Mathematics, Russian, and German. It is noted that:

(1) Exactly 11 students have a grade of 2 in Mathematics.

(2) Exactly 19 students have a grade of 2 in Russian.

(3) Exactly 23 students have a grade of 2 in German.

(4) Exactly one student has a grade of 2 in all three subjects.

(5) Exactly 4 students have a grade of 2 in Mathematics and German, but not in Russian.

(6) Exactly 7 students have a grade of 2 in Russian and German, but not in Mathematics.

(7) Exactly 2 students have a grade of 2 in Mathematics and Russian, but not in German.

Determine from these statements how many students in this group have a grade of 2 in exactly one and how many in none of the specified subjects!
From (1), (4), (5), and (7), it follows that because $11-1-4-2=4$, exactly 4 students have a grade of 2 only in Mathematics.

From (2), (4), (6), and (7), it follows that because $19-1-7-2=9$, exactly 9 students have a grade of 2 only in Russian.

From (3), (4), (5), and (6), it follows that because $23-1-4-7=11$, exactly 11 students have a grade of 2 only in German.

Because $4+9+11=24$, exactly 24 students in this group have a grade of 2 in exactly one of the mentioned subjects.

Since exactly 13 students have a grade of 2 in exactly two of the subjects according to (5), (6), and (7), and one more student is added according to (4), it follows that $24+13+1=38$ students have a grade of 2 in at least one of the subjects. Therefore, exactly one student in this group has a grade of 2 in none of the mentioned subjects.

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