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Of a group of 200 persons, 100 are interested in Music, 70 are interested in photography and 40 like skiing. Also 40 are interested in both Music and photography, 30 ni both Music and skiing, 20 ni both photography and skiing, while 20 are interested in photography but not in Music or skiing How many persons are interested in al the three?
A. 40 B. 20 C. 5 D. 8 E. 10

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Réponse:

To find the number of persons interested in all three activities (Music, photography, and skiing), we can use the principle of inclusion-exclusion.

Let:

- A = number of persons interested in Music

- B = number of persons interested in photography

- C = number of persons interested in skiing

- n(A) = 100

- n(B) = 70

- n(C) = 40

- n(A ∩ B) = 40

- n(A ∩ C) = 30

- n(B ∩ C) = 20

- n(B - A - C) = 20

We want to find n(A ∩ B ∩ C).

Using the principle of inclusion-exclusion:

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C) + n(B - A - C)

Plugging in the values:

200 = 100 + 70 + 40 - 40 - 30 - 20 + n(A ∩ B ∩ C) + 20

200 = 220 - 90 + n(A ∩ B ∩ C)

n(A ∩ B ∩ C) = 70

Therefore, there are 10 persons interested in all three activities: Music, photography, and skiing.

Answer: E. 10

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