Blood tests of 100 people showed that 56 had the \( A \) antigen and 42 had the \( B \) antigen. Another 26 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?

To determine how many people are of type \( AB \) (having both \( A \) and \( B \) antigens), we can use a bit of set theory and basic arithmetic based on the information given. Let's start by summarizing the information: - Total people tested: 100 - People with \( A \) antigen: 56 - People with \( B \) antigen: 42 - People with neither antigen (type \( O \)): 26 From the total of 100 people, 74 people either have \( A \), \( B \), or both \( A \) and \( B \) antigens because: \[ 100 - 26 = 74 \] Now, let's denote: - \( n(A) \) = number of people with antigen \( A \) = 56 - \( n(B) \) = number of people with antigen \( B \) = 42 - \( n(A \cap B) \) = number of people with both antigens \( A \) and \( B \) Using the principle of inclusion-exclusion, we can calculate the number of people with either \( A \) or \( B \) or both: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] Since \( n(A \cup B) = 74 \), we substitute: \[ 74 = 56 + 42 - n(A \cap B) \] Combining and rearranging gives us: \[ 74 = 98 - n(A \cap B) \] \[ n(A \cap B) = 98 - 74 \] \[ n(A \cap B) = 24 \] So, there are **24 people of type \( AB \)**, having both the \( A \) and \( B \) antigens. For the Venn diagram, you would draw two overlapping circles: 1. The left circle represents blood type \( A \), containing 32 (56 total with antigen \( A \) minus 24 for \( AB \)). 2. The right circle represents blood type \( B \), containing 18 (42 total with antigen \( B \) minus 24 for \( AB \)). 3. The intersection of the two circles will hold 24 (type \( AB \)). 4. Outside the circles, you would label 26 people as having blood type \( O \). This clearly shows the distribution of blood types among the 100 people tested!