Question

Из 80 учеников девятых классов в олимпиаде по физике участвовали 41 человек, по химии 39 человек, по математике 41. В олимпиадах по физ и хим принимали участие 75 человек, по физ и мат 62 человека, по хим и мат 59 человек. Сколько учеников не участвовали ни в одной олимпиаде, если во всех трех участвовали только трое?

Answer 1

2 человека не участвовали в олимпиадах.

Answer 2

Solution

To solve this problem, we can use the principle of inclusion-exclusion. We'll start by finding the total number of students who participated in at least one Olympiad. Then, we'll subtract the number of students who participated in exactly one Olympiad, add back the number of students who participated in exactly two Olympiads, and finally subtract the number of students who participated in all three Olympiads.

Let's calculate each step:

1. Total students participating in at least one Olympiad: - Physics (P) = 41 - Chemistry (C) = 39 - Mathematics (M) = 41 - Participated in at least one Olympiad = P + C + M - (P ∩ C) - (C ∩ M) - (M ∩ P) + (P ∩ C ∩ M)

2. Students participating in exactly one Olympiad: - Participated in exactly one Olympiad = (P - (P ∩ C) - (P ∩ M)) + (C - (C ∩ M) - (C ∩ P)) + (M - (M ∩ P) - (M ∩ C))

3. Students participating in exactly two Olympiads: - Participated in exactly two Olympiads = (P ∩ C) + (C ∩ M) + (M ∩ P)

4. Students participating in all three Olympiads: - Participated in all three Olympiads = P ∩ C ∩ M

Finally, we'll calculate the number of students who did not participate in any Olympiad using the principle of inclusion-exclusion.

Let's proceed with the calculations.