How Many Times Can 16 People Be Put in Groups of 4 Without Being With the Same Person Twice?

When organizing activities involving 16 individuals, it's crucial to determine the optimal way to group them into groups of 4 without repeating any pairings. This problem can be analyzed using combinatorial reasoning and concepts from group theory.

Total Group Combinations

First, let's calculate the total number of groups that can be formed from 16 people, where each group consists of 4 people. We use the combination formula to calculate the number of ways to choose 4 people from a group of 16:

[ binom{16}{4} frac{16!}{4!(16-4)!} frac{16 times 15 times 14 times 13}{4 times 3 times 2 times 1} 1820 ]

This means there are 1,820 different ways to choose 4 people from a group of 16.

Pairing Constraints

To ensure no two individuals are paired together more than once, we can model the problem using a graph where each person is a vertex, and each group of 4 people forms edges between the vertices. This setup helps us understand the constraints of the problem.

Finding the Maximum Number of Groupings

The maximum number of times you can create groups of 4 without repeating pairs can be calculated using the concept of a Steiner system or balanced incomplete block design. A Steiner system (S(t,k,n)) is a collection of k-subsets (blocks) of an n-set such that each t-subset of the n-set is contained in exactly one block. In this case, we are dealing with an S(2,4,16) system (where t2, k4, and n16).

The theoretical maximum for this setup is given by the formula:

[ text{Number of groups} frac{n(n-1)}{k(k-1)} ]

Where:

n is the total number of people (16), and k is the group size (4).

[ text{Number of groups} frac{16 times 15}{4 times 3} frac{240}{12} 20 ]

Therefore, 16 people can be grouped into groups of 4 a maximum of 20 times without having the same pair of people in the same group more than once.

Examples of Groupings

Here are 5 examples of how 16 people can be grouped into 4-person clusters without repeating any pair:

ABCD, EFGH, IJKL, MNOP AEIM, BFJN, CGKO, DHLP AFKP, BGLM, CHEI, DFIN AGIO, BHJP, CFKM, DELO AHKN, BELO, CFIK, DGJM

These examples demonstrate a valid way to group individuals where no pair appears more than once. Grouping individuals without repeating pairings is a practical application of combinatorial theory and is widely used in various scenarios such as team-building exercises, sports tournaments, and conference planning.