Answer:
To find the number of candidates who offer all three subjects (mathematics, history, and biology), you can use the principle of inclusion-exclusion. Start by adding up the number of candidates who offer each subject:
Explanation:
Biology (B) = 52
History (H) = 60
Mathematics (M) = 96
Now, you know that:
16 offer both mathematics and history (M ∩ H).
21 offer biology and history (B ∩ H).
22 offer mathematics and biology (M ∩ B).
Now, use the principle of inclusion-exclusion to find the total number of candidates who offer at least one of the three subjects:
Total = B + H + M - (M ∩ H) - (B ∩ H) - (M ∩ B) + (M ∩ H ∩ B)
Total = 52 + 60 + 96 - 16 - 21 - 22 + (M ∩ H ∩ B)
Now, calculate the total using the numbers provided:
Total = 52 + 60 + 96 - 16 - 21 - 22 + (M ∩ H ∩ B)
Total = 249 + (M ∩ H ∩ B)
Now, to find the number of candidates who offer all three subjects (M ∩ H ∩ B), you need to subtract the total (249) from the sum of those who offered each subject:
M ∩ H ∩ B = Total - (B + H + M - (M ∩ H) - (B ∩ H) - (M ∩ B))
M ∩ H ∩ B = 249 - (52 + 60 + 96 - 16 - 21 - 22)
M ∩ H ∩ B = 249 - 139
M ∩ H ∩ B = 110
So, there are 110 candidates who offer all three subjects: mathematics, history, and biology.