Blood tests of 100 people showed that 65 had the $$ A $$ antigen and 36 had the $$ B $$ antigen. Another 25 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, AB, and O. How many people are of type AB, having both the A and B antigens? H

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We start by letting $$ x $$ be the number of people who have both the $$ A $$ and $$ B $$ antigens (type $$ AB $$). According to the inclusion‐exclusion principle, the total number of people with at least one antigen is given by

$$
\text{Number with $$A$$ or $$B$$} = \text{Number with $$A$$} + \text{Number with $$B$$} - \text{Number with both}
$$

From the problem, 25 people had neither antigen, so the number of people with at least one antigen is

$$
100 - 25 = 75
$$

Thus, we have

$$
65 + 36 - x = 75
$$

Solving for $$ x $$:

$$
101 - x = 75 \quad \Longrightarrow \quad x = 101 - 75 = 26
$$

So, there are $$ 26 $$ people of type $$ AB $$.

Now we can determine the numbers for the other types:

- **Type $$ A $$ only:** $$ 65 - 26 = 39 $$ people.
- **Type $$ B $$ only:** $$ 36 - 26 = 10 $$ people.
- **Type $$ O $$:** $$ 25 $$ people (given).

Below is a Venn diagram representing the situation:

```markdown
             _______       
            /       \      
           /    A    \     
          / 39      26\    
         /             \   
        /_______________\  
          \           /        
           \    B    /         
            \ 10   /           
             \____/            
             
Outside both circles: 25 (Type O)
```

- The left circle represents people with the $$ A $$ antigen.
- The right circle represents people with the $$ B $$ antigen.
- The intersection (overlap) has $$ 26 $$ people with both antigens (type $$ AB $$).
- The area exclusive to the left circle has $$ 39 $$ people (type $$ A $$ only), and the area exclusive to the right circle has $$ 10 $$ people (type $$ B $$ only).
- Outside both circles are the $$ 25 $$ people with neither antigen (type $$ O $$).

Thus, the number of people of type $$ AB $$ is $$ \boxed{26} $$.
There are 26 people of type AB, having both the A and B antigens.

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