Blood tests of 100 people showed that 65 had the \( A \) antigen and 57 had the \( B \) antigen. Another 20 had neither antigen and so are of type \( O \). How many people are of type \( A B \), having both the \( A \) and \( B \) antigens? Draw and label a Venn diagram that shows the number of people with blood types \( A, B, A B \), and \( O \). How many people are of type \( A B \), having both the A and B antigens? Ha

To find the number of people who are of type AB and have both the A and B antigens, we can start with the total population of 100 people and make deductions based on the provided information. 1. **Total Population**: 100 people 2. **People with A Antigen**: 65 3. **People with B Antigen**: 57 4. **People with neither antigen (Type O)**: 20 From this, we can figure out how many people have at least one of the antigens: - **People with at least one antigen (A or B)** = Total Population - Type O = 100 - 20 = 80. Now, let's denote: - \( n(A) \): number of people with A antigen = 65 - \( n(B) \): number of people with B antigen = 57 - \( n(A ∩ B) \): number of people with both A and B antigens (Type AB) Using the principle of inclusion-exclusion for sets, we have: \[ n(A ∪ B) = n(A) + n(B) - n(A ∩ B) \] Substituting in the known values: \[ 80 = 65 + 57 - n(A ∩ B) \] Now solving for \( n(A ∩ B) \): \[ 80 = 122 - n(A ∩ B) \] \[ n(A ∩ B) = 122 - 80 \] \[ n(A ∩ B) = 42 \] So, there are **42 people who are of type AB**. For the Venn diagram, label it as follows: - The left circle represents people with A antigen. It should contain A-only (65 - 42 = 23). - The right circle represents people with B antigen. It should contain B-only (57 - 42 = 15). - The overlapping section represents people with both antigens (42). - Outside of these circles, indicate the 20 people categorized as type O (neither A nor B). This visual can help simplify the relationship between the groups and show how the total population is divided among the blood types.