Final answer:
To determine the number of students who studied all three Nigerian languages, we applied the inclusion-exclusion principle. After using the given values for students studying Hausa, Igbo, Yoruba, and pairwise combinations, we find that 8 students studied all three languages.
Step-by-step explanation:
To find the number of students who studied all three Nigerian languages (Hausa, Igbo, and Yoruba), we use the principle of inclusion-exclusion for three sets. This principle is used to avoid overcounting the number of students when some of them are counted more than once because they take more than one language. The formula for the principle is:
N(H ∪ I ∪ Y) = N(H) + N(I) + N(Y) - N(H ∩ I) - N(H ∩ Y) - N(I ∩ Y) + N(H ∩ I ∩ Y)
Where:
- N(H ∪ I ∪ Y) is the number of students taking at least one language.
- N(H), N(I), and N(Y) are the number of students taking Hausa, Igbo, and Yoruba, respectively.
- N(H ∩ I), N(H ∩ Y), and N(I ∩ Y) are the number of students taking each pair of languages.
- N(H ∩ I ∩ Y) is the number of students taking all three languages, which is what we want to find.
We are given:
- 100 students take at least one language.
- 65 students take Hausa (H).
- 45 students take Igbo (I).
- 42 students take Yoruba (Y).
- 20 students take both Hausa and Igbo (H ∩ I).
- 25 students take both Hausa and Yoruba (H ∩ Y).
- 15 students take both Igbo and Yoruba (I ∩ Y).
By substituting these values into the formula we get:
N(H ∩ I ∩ Y) = 100 - (65 + 45 + 42) + 20 + 25 + 15
N(H ∩ I ∩ Y) = 100 - 152 + 60
N(H ∩ I ∩ Y) = 8
Therefore, 8 students studied all three languages.