Mathematics: Factorials

Factorials
The factorial is a function that can be performed on any non-negative integer. It is represented by the ! sign written after the integer on which it is being performed. The factorial of an integer is the product of all positive integers less than or equal to the number. For example, 4! (read '4 factorial') is calculated as
.

Since 0 is not itself a positive integer, nor does it have any positive integers less than it, 0! cannot be calculated using this method. Instead, 0! is defined by convention to equal 1. This makes sense if you consider the pattern of descending factorials:












Permutations
For any given set of data, the individual elements in the set may be arranged in different groups containing different numbers of elements arranged in different orders. For example, given the set of integers from one to three, inclusive, the elements of the set are 1, 2, and 3: written as {1, 2, 3}. They may be arranged as follows: 1, 2, 3, 12, 21, 13, 31, 23, 32, 123, 132, 213, 231, 312, and 321. These ordered sequences of elements from the given set of data are called permutations. It is important to note that in permutations, the order of the elements in the sequence is important. The sequence 123 is not the same as the sequence 213. Also, no element in the given set may be used more times as an element in a permutation than it appears as an element in the original set. For example, 223 is not a permutation in the above example because the number 2 only appears one time in the given set.

To find the number of permutations of r items from a set of n items, use the formula
. When using this formula, each element of r must be unique. Also, this assumes that different arrangements of the same set of elements yields different outcomes.

For example, 123 is not the same as 321; order is important.

A special case arises while finding the number of possible permutations of n items from a set of n items. Because
, the equation for the number of permutations becomes simply
 The same result is true for
. Both of these cases are a result of the fact that 0! and 1! are both equal to 1.
If a set contains one or more groups of indistinguishable or interchangeable elements (e.g., the set {1, 2, 3, 3}, which has a group of two indistinguishable 3's), there is a different formula for finding distinct permutations of all n elements.

Use the formula
is the number of permutations, <i>n</i> is the total number of elements in the set, and <br><img data-cke-saved-src=" /> are the number of identical elements in each group (e.g., for the set {1, 1, 2, 2, 2, 3, 3},
,
, and
).

It is important to note that each repeated number is counted as its own element for the purpose of defining n (e.g., for the set {1, 1, 2, 2, 2, 3, 3},
, not 3).

To find the number of possible permutations of any number of elements in a set of unique elements, you must apply the permutation formulas multiple times. For example, to find the total number of possible permutations of the set {1, 2, 3} first apply the permutation formula for situations where
is 3 and <i>r</i> is 2.<br><img data-cke-saved-src=" />

To find the number of permutations when one element is used, use the formula
is 3 and <i>r</i> is 1.<br><img data-cke-saved-src=" />

Find the sum of the three formulas:
 total possible permutations.
Alternatively, the general formula for total possible permutations can be written as follows:


Combinations
Combinations are essentially defined as permutations where the order in which the elements appear does not matter. Going back to the earlier example of the set {1, 2, 3}, the possible combinations that can be made from that set are 1, 2, 3, 12, 13, 23, and 123.
In a set containing n elements, the number of combinations of r items from the set can be found using the formula
. Notice the similarity to the formula for permutations. In effect, you are dividing the number of permutations by r! to get the number of combinations, and the formula may be written . When finding the number of combinations, it is important to remember that the elements in the set must be unique (i.e., there must not be any duplicate items), and that no item may be used more than once in any given sequence.


Practice
P1. Ichiro has 4 shirts, 1 jacket, and 5 different pairs of pants that he packed for his work trip. If wearing a jacket is optional, how many outfit combinations can he make?
P2. Determine the number of permutations and combinations of the following:
Choose 3 from the set:

(b) Choose 2 from the set:

(c) Choose 4 from the set:


P1. To start with, each shirt can be matched with each pair of pants, so that would be
 combinations.
Since the jacket is optional each of the 20 can be either with or without the jacket, i.e.
.

There are 40 distinct combinations he could wear.
P2. (a)

(b)

(c)