Permutations

Author: Will Bickford
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Definitions

The following (implicit) definition of a permutation is taken from page 321 of
Discrete Mathematics and Its Applications, 5th ed. (Kenneth H. Rosen):

Definition: The number of ways in which n objects can be arranged n ways is n!.

When working with permutations, it is useful to define the permutation function. The permutation function provides a method to calculate the number of ways in which a set of n distinct objects can be uniquely arranged into a set of r objects. The explicit definition of the permutation function is given in Figure 1 below. As you can see, the permutation function is constructed using the factorial function.

Permutation Function:
Figure 1 - The permutation function defined in terms of the factorial function.

The idea behind the factorial function is simple. For a given integer, n, one recursively multiplies n by n-1 by n-2 ... by 1. Loosely stated: Start with an integer and multiply it by every integer between that integer and 1 inclusive.

In order to effectively deal with factorials there must be a few definitions. Presented in Figure 2, it is defined that 0! = 1! = 1. The reason for setting 0! = 1 is simple. The permutation function must satisfy P(n,n) = n! because, by definition, a permutation of n objects from a set of n objects is n!. Closer inspection of P(n,n) shows that P(n,n) = n!/(n-n)! = n!/0!. If 0! were undefined the permutation function would not generalize to the most general case!

Factorial Function:
A useful property:
Figure 2 - The factorial function is defined recursively. By definition, 0! = 1! = 1.

A "Real-World" Example

Usually an example is extremely useful in describing the nature of any new concept, and permutations are no different. Suppose one has 3 ducks and that one also has 3 ponds. The ducks are distinct. There is a red duck, a blue duck, and a green duck. Ducks will only live peacefully on a pond with ducks of the same color. Color is the only defining characteristic of a duck. One specifically requires that all ducks live in relative peace because the cost of a duck burial is prohibitively expensive. Also, the caretaker is particularly fond of one's ducks - so losing a duck would adversely affect the emotional state of the caretaker.

Figure 3 - The 3 ducks and the 3 ponds.

Granted, ducks of the variety described above are hard to come by, but they suit the example perfectly. Notice that there are 3 distinct objects which we desire to order into 3 ponds. For the purposes of this example, the ponds will be numbered 1, 2, and 3. I have used easy-to-define characteristics simply to illustrate the issues at hand. In reality we would analyze the situation using more realistic comparisons.

There are two ways to approach this problem: 1) The HARD way, and 2) the EASY way. The HARD way involves explicitly counting all possible permutations of the duck-pond system. The EASY way is to simply apply the permutation function: P(3,3) = 3! = 6. Let's see how the HARD way corresponds to the EASY way.

The first pond, Pond #1, can house any duck. This provides 3 options for choosing a duck for Pond #1. Once a duck has been chosen for Pond #1, there are 2 remaining ducks for Pond #2. This provides 2 options for choosing a duck for Pond #2. The final pond, Pond #3, can only house the remaining duck. This provides only 1 option for choosing a duck for Pond #3. The multiplication rule is used here because we are choosing the number of ways in which a sequence of events can occur (choosing a duck for a pond). The events are connected because putting a red duck in Pond #1 and then a blue duck in Pond #2 is DIFFERENT from putting a blue duck in Pond #1 and then a red duck in Pond #2.

Comparing the two methods:
Figure 4 - Equating the EASY way and the HARD way.

Putting It All to Good Use

Now that you've studied permutations and followed an example problem, it's time to flex your mental muscle and solve a problem on your own!

The duck population has ballooned! There are now 10 distinct ducks and 10 ponds (the caretaker has found 7 new ponds to accommodate the new ducks). Your task is to determine the number of ways in which the ducks can be put into the ponds. As before, color is the distinguishing characteristic (there is now 1 red, 1 blue, 1 green, 1 yellow, 1 black, 1 white, 1 cyan, 1 magenta, 1 grey, and 1 invisible duck. The invisible duck is particularly dangerous for the caretaker because only the invisible duck's wake can be seen. The caretaker has taken precautions so that the pond which the invisible duck is put into will be secured.

As before, ducks will only live peacefully on a pond with ducks of the same color. Assume that the invisible duck is a duck with the color invisible. Obviously, then, you are tasked with sorting 10 distinct ducks into 10 ordered ponds.

Do not try to solve this problem explicitly!

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Based upon material from Discrete Mathematics and Its Applications, 5th ed. (Kenneth H. Rosen).

The duck model is based upon a tutorial written for the Rhinoceros 3D training manual.