Counting Ways to Form 4 Groups of 4 People from 16: A Combination and Permutation Exploration

In this article, we explore a classic combinatorics problem: How many distinct groups of 4 people can be formed from a total of 16 people when the groups are distinct? We will use both combinations and permutations to solve this problem and introduce the partition counting problem as a more general method.

Introduction to Combinations and Permutations

Combinations and permutations are fundamental concepts in combinatorics, a branch of mathematics that deals with the study of finite or countable discrete structures. When solving problems related to grouping or distributing objects, these concepts are crucial.

Forming 4 Groups of 4 People from 16

Imagine you have a group of 16 people and you need to form 4 distinct groups, each containing 4 people. We can use the combination formula to calculate the number of ways to do this step-by-step.

Choosing the First Group

To begin, we choose the first group of 4 people from the 16 available. The number of ways to do this is given by the combination formula:

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

Choosing the Second Group

After choosing the first group, 12 people remain. The number of ways to choose the second group of 4 people from these 12 is:

[ binom{12}{4} frac{12!}{4!(12-4)!} frac{12 times 11 times 10 times 9}{4 times 3 times 2 times 1} 495 ]

Choosing the Third Group

Now, 8 people remain. The number of ways to choose the third group of 4 people from these 8 is:

[ binom{8}{4} frac{8!}{4!(8-4)!} frac{8 times 7 times 6 times 5}{4 times 3 times 2 times 1} 70 ]

Choosing the Fourth Group

Finally, 4 people remain and we can choose all of them to form the fourth group in only one way:

[ binom{4}{4} 1 ]

Total Number of Ways

Since the groups are distinct, we multiply the number of ways to choose each group together:

[ text{Total ways} binom{16}{4} times binom{12}{4} times binom{8}{4} times binom{4}{4} ]

Substituting the values calculated:

[ text{Total ways} 1820 times 495 times 70 times 1 ]

Calculating this step-by-step:

[ 1820 times 495 901900 ] [ 901900 times 70 63133000 ]

Thus, the total number of ways to form 4 distinct groups of 4 people from 16 people is:

[ 63133000 ]

Partition Counting Problem

This is what I like to call the partition counting problem. The general statement is:

How many ways can you partition N objects into k groups with n_k members of each group?

Note that partitioning implies no leftover members from the N or the sum of the n_k is N.

Visualizing the Process

The solution is easiest to see when you visualize the process of partitioning the objects in a convenient way. Imagine a set of N slots with barriers added between the slots to represent the boundaries between the partitions you will fill.

Example with 3 Objects and 2 Partitions

For instance, with 3 objects and 2 partitions, it might look something like this:

A1 A2 B1

Where partition A has two members and partition B has just one. If the elements of the original set are {1, 2, 3}, we can imagine the following ways to order the elements:

1 2 3 2 1 3 1 3 2 3 1 2 2 3 1 3 2 1

Since we don't care about the order of elements within a given partition but we do care about the order of the specific partitions, it is clear that we have over-counted. Specifically, for each partition, there are n_k! ways to arrange the elements, and since there are n_k partitions, we have over-counted by the product of all the n_k!.

General Formula

The general formula is:

[ P_{N}^{n} frac{N!}{prod_{k1}^{n} n_k!} ]

Where P_{N}^{n} is the number of ways to partition N elements into the partitions with sizes denoted by the vector n.

Applying the Formula

For the original question of partitioning 16 elements into 4 groups of 4 elements, the number of ways to do this is given by:

[ P_{16}^{[4,4,4,4]} frac{16!}{4!^4} 63063000 ]

This formula provides a more elegant and general solution to the partition counting problem.