II. Exponents (Exponential Sets or Saliusian Sets;

III. Permutations (Factorial Sets;

IV. Arrangements Sets (

V. Combinations (

VI. Sets of Letters, Words, Names

VII. Generating Words from Letters, Passwords: Lexicographically and Randomly

VIII. Resources in Theory of Probability, Mathematics, Combinatorics, Permutations, Software

**PermuteCombine** is the summit of all mathematical generation. The program calculates and generates exponents, permutations, arrangements, and combinations for any numbers and words. The sets can be sequential (lexicographic order) or random.

We saw that the * probability theory* is founded on

It is obvious by now that probability always deals with *discrete* elements; i.e. elements we can always count. And thus our main task is to take inventory of all elements in the set and then determine the group of favorable cases. In most cases, such a task is very easy. We can easily count the faces of a die: 6. Next, we only have to state what probability we want to calculate. Let's take the easiest example: The probability to get any point-face (e.g. 5) in one roll of the die. There is exactly one (1) point-face reading 5. Therefore, the probability to get a 5 in one roll of the dice is *1/6*, or *1 in 6*, or *.166666666…*

That easy situation of enumeration is quite rare in the grand scheme of things. In many cases, the total number of possible cases can only be calculated by * combinatorics*, a branch of mathematics.

Any finite number of elements can be put together in groups based on certain rules. Such groups are known as sets. The sets can comprise from 0 elements to infinity (actually, a huge, gigantic, cosmic amount of finite elements). The *amount* of elements in the set is always an integer number, although the elements themselves can be fractions, decimal numbers, and even transcendental numbers. For example, sale volumes over the past week can read something like {23,457.55, 17,444.02, 8,995.34, 7,234.75, 21,567.99, 9,999.99, 11,234.55}. There are exactly 7 total elements in the set. What is the probability to *guesstimate* a sale over 10,000? There are 3 favorable cases; p = 3/7.

There is a lot of confusion in the field of sets, both in the academia and among laypersons. The *combination* and *permutation *are the most commonly used terms. Worse, people interchange *combination* and *permutation* without any restriction! Apparently, in the British system of education, *combination* and *permutation* are synonyms! We must establish order and discipline in this classroom! (Don't worry lads! I am mild-mannered!)

**There are four distinct types of sets, from the most inclusive to the least inclusive: Exponents, permutations, arrangements, combinations.**

The *exponents* represent the most inclusive of the four types of sets. The *exponent set* includes *permutations*; the *permutation set* includes *arrangements*; the *arrangement set* is inclusive of *combinations*.

The number sets (numerical or numeric) are the most important mathematically. We can substitute the numbers by alphanumerical elements, such as words, names, any strings of characters. In the case of the alphanumerical sets, mathematics works with the *indices* (indexes) of the respective elements.

An example of * exponents* (N=3, M=3): 111,112,113,121,122,123,131,132, etc. (a total of 27 sets). The best-known examples are the drawings of

An example of * permutations* (for N = 3): 1 2 3, 1 3 2, 2 1 3, 2 3 1, 3 1 2, 3 2 1 (6 elements: 1* 2 * 3)). There aren't many practical examples for permutations.

An example of * arrangements* (for N = 3, M = 2): 1 2, 1 3, 2 1, 2 3, 3 1, 3 2. (6 elements in set: 3 * 2).The best-known examples are the trifectas, the results of horse races (trifectas in the United States, triactors in Canada, top-three-finishers in Britain, etc.)

An example of * combinations* (for N = 3, M = 2): 1 2, 1 3, 2 3 (3 elements: 3 * 2 / 1 * 2). The best-known examples are the drawings of

The exponents are very important. They are capable of solving a wide range of probability problems. Since the exponents accept both unique elements and duplicates (repeat-elements), they can solve problems of gigantic proportions and importance. For instance: Can intelligent life, as present on earth, have a duplicate anywhere in the Gigantic Universe? We will tackle the issue later in this book.

The pick-3 or pick-4 lottery games are the most commonly known examples of exponents. Each digit of the pick games takes values between 0 and 9 (10 values). The pick-3 game has a total of 10 to the power of 3 (10 ^ 3) = 1000 combinations. Examples: 013 (equivalent to 0,1,3 or 0-1-3)

The soccer pools, such as totocalcio (in Italy), have 3 outcomes for 13 games; 3 to the power of 13 (3 ^ 13) = 1,594,323 possibilities (total possible cases). Examples: 1,X,2,X,X,2,2,1,1,1,X,1,2 or 1X2XX22111X12.

The parameters for soccer pools (known as totocalcio, pronosport or other names from nation to nation) are: 13 (items per set), 0 (lower bound), 2 (upper bound). 1 represents home victory, 2 is for win for the visitor, and 0 (marked as X on play slips) stands for a tie. The pick-3, pick-4 games have a lower bound of 0, and an upper bound of 9. Items per set: 3 (in pick-3) and 4 (in pick-4).

The formula of exponents is:

**Exponent (N, M) = N ^ M = N ^{M}**

(^ represents the *raising to power* operator; they also use the superscript as indicator of *raising to power or exponent, exponential operator*).

The lottery in the State of Pennsylvania has a digit game named Quinto. It draws 5 digits (**M** = 5), from 0 to 9 (a total of ten elements; **N** = 10). **10 ^{5} = 100,000**.

The exponents (exponential sets) grow much faster than anything else, including permutations!

The factorials also grow extremely rapidly, but with a lesser intensity compared to the exponents.

The formula of permutations is:

**Permutation (N) = N! = 1 x 2 x 3 x … x N**

As I said previously, it's hard to find real-life examples of permutations. We can visualize an example by taking all the playing cards in a suit. There are 13 cards, from 2 to A: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A. We can arrange the 13 playing cards in a variety of orders, without removing any card. Factorial of 13 or 13! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 = 6,227,020,800 (more than 6 billion ways!) The first way is *2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A*; the next sequence is *2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, A, K*; the 3rd sequence is *2, 3, 4, 5, 6, 7, 8, 9, 10, J, A, Q, K*; etc., etc.

**n P m = n! / (n - m)!**

**n! = 1 x 2 x 3 x...x (n-m) x (n - m + 1) x (n – m + 2) x...x n**

**(n - m)! = 1 x 2 x 3 x...x (n - m)**

We can simplify the fraction **n! / (n - m)!** and obtain:

**(n – m + 1) x (n – m + 2) x ... x n**

It is the factorial of a series of elements beginning at the term **n – m + 1** and ending at the term **n**. That's the formula of the arrangements:

**Arrangements (N, M) = N x (N – 1) x (N – 2) x (N – 3) x ... x (N – M + 1)**

The *exactas* (top two finishers), or *trifectas* (top three finishers), or *superfectas* (top four finishers) in horse racing are some of the most common representations of the arrangements. If the famous horse race known as The Kentucky Derby has 20 horses, the total possible amount of straight trifectas is:
**A (20, 3) = 20 x 19 x 18 = 6840**.

The boxed trifectas represent, in fact, the * combinations* of (

The * combinations* are the equivalent of

The combinations formula is:

**Combinations (N, M) = Arrangements (N, M) / Permutations (M) = {N x (N – 1) x (N – 2) x (N – 3) x … x (N – M + 1)} / {1 x 2 x 3 x … x M}**

As in this example of the most common lotto game in the world: 6 from 49:

Total possible 6/49 combinations = (49 x 48 x 47 x 46 x 45) / (1 x 2 x 3 x 4 x 5 x 6) = 13,983,816.

The * combination set* is, by far, the most extensively researched. I reckon it is so because of the lotto games! Even members of the academia want to get rich. No kidding! A group of professors and staff members at Bradford University and College in Britain won the lotto jackpot! It happened on October 21, 2006 in the United Kingdom National Lottery. They played a lotto strategy I started in the early 1990s. I presented that

There are special lottery games: Powerball, Mega Millions, Euromillions. They are most accurately defined as *two-in-one* games. The numbers drawn from the second chamber can be equal to any of the numbers drawn in the first set. Here is a screenshot for the Powerball / Mega Millions game:

And here is a screenshot for the Euromillions game played in several European countries:

*Permutations*

*permutation, sequential*

*random*

*lexicographic*

*Lexicographical, Order,*

*combination*

Here are fragments of the output file for the *word-generating* option for * Combinations* in

- * Combinations*,

to...

The word generation can be used at horse races to generate sets of horse *names*, instead of *post position numbers*. This feature is useful in sports betting, too. A user can write, for example, all the NFL teams in a list file. Then generate parlays every week, using the feature: * Arrangements: Words, Random*. Select, for example, 5 words (teams) per set. Check the output and disregard the sets with teams from the same game for that week.

The permutation, exponential, and combination word features can work as well. No word generation for Powerball, Mega Millions, Euromillions — not of much use, in my mind.

Suppose we write the 26 letters of the English alphabet in a 26-line text file, one letter per line. We can generate an astronomical number of permutations (words). But the permutations have a fixed length (number of letters), whereas the natural languages have words of a variety of lengths. The exponents serve the purpose better. The words vary from one letter to some teens. The first letter of a word can be from A to Z; the second letter of a word can be from A to Z; the third letter of a word can be from A to Z; etc. The words can have repeat letters.

I wrote another computer program that generates random words in a manner close to the natural languages. The words can have repeat letters; the words are of variable length; a word must contain at least one vowel. The combinatorics software is named **Writer**.

- Please be advised that the permutations can grow astronomically! A low number such as 10 has a factorial of 3,628,800 (total permutations)! Factorial of 100 (100!): Don't even try! It's mind blowing!
- Please be advised that the arrangements, too, can grow astronomically!
- Try first random numbers to see how many sets there are. The default number of sets to generate randomly is 100. The user, of course, can choose different amounts to generate.

Said Tabaki Paravicius, netizen extraordinaire: *"You ain't seen nothing quite like it, Kulai! PermuteCombine is the mother of all generations."*

Read Ion Saliu's first book in print: *Probability Theory, Live!*

~ Founded on valuable mathematical discoveries with a wide range of scientific applications, including the organic connection between probability theory and sets of numbers — permutations, combinations, arrangements, Ion Saliu's sets.

- Download the
**permutation, combination, probability software**:

•, the universal permutations, arrangements and combinations generator for any numbers and words;**Permute Combine**

•, the universal permutations, arrangements and combinations lexicographic indexing (ranking);**LexicographicSets**

•, the universal combinations generator for any lotto, Keno, Powerball game: N numbers taken M at a time, in K steps;**Combinations**

•, random generator of letters to words, passwords, sentences;**WRITER**

•~ lexicographic indexing superseded by**NthIndex**;*LexicographicSets*

•- highly automated lexicographical indexing for lotto draw files.**DrawIndex**

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