A person wishes to make up as many different parties as he can out of his 16 friends such that each partu consists of the same number of persons. How many friends should be invite ?

Answer

To solve the problem of how many friends a person should invite to make as many different parties as possible, where each party consists of the same number of persons, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Friends**:
The total number of friends available is given as 16.

2. **Understanding the Problem**:
We need to form parties such that each party has the same number of friends. This means we need to find a number \( r \) (the number of friends in each party) that allows us to maximize the number of different combinations of parties.

3. **Using Combinations**:
The number of ways to choose \( r \) friends from \( n \) friends is given by the combination formula \( \binom{n}{r} \). We want to maximize this value.

4. **Maximizing Combinations**:
It is a known fact that the value of \( \binom{n}{r} \) is maximized when \( r \) is equal to \( \frac{n}{2} \). Since \( n = 16 \), we calculate:
\[ r = \frac{16}{2} = 8 \]

5. **Conclusion**:
Therefore, to maximize the number of different parties, the person should invite 8 friends to each party.

### Final Answer:
The person should invite **8 friends** to the party.