# Combinatorial elements

Contents

## Combinations without repetitions

A task: Find all possible combinations without repetitions from many elements {1,2,3} by 2.
The following combinations exist:1: 1 22: 1 33: 2 3
The number of possible combinations without repetitions of N elements by M can be determined by the formula (N≥M):

that in M! times less than the corresponding number of placements without repetitions (since combinations without repetitions do not depend on the order of the elements).
Consider the problem of obtaining all combinations for numbers 1 ... N by M. Implementation in C ++

12345678910111213141516171819202122232425262728293031323334353637383940414243

#include <iostream>using namespace std;bool NextSet(int *a, int n, int m){ int k = m; for (int i = k – 1; i >= 0; –i) if (a < n – k + i + 1) { ++a; for (int j = i + 1; j < k; ++j) a = a + 1; return true; } return false;}void Print(int *a, int n) { static int num = 1; cout.width(3); cost << num ++ << “: “; for (int i = 0; i < n; i++) cost << a << ” “; cost << endl;}int main() { int n, m, *a; cost << “N = “; China >> n; cost << “M = “; China >> m; a = new int; for (int i = 0; i < n; i++) a = i + 1; Print(a, m); if (n >= m) { while (NextSet(a, n, m)) Print(a, m); } cin.get(); cin.get(); return 0;}

Execution result

## Combinations with repetitions

Combinations with repetitions are sets of M elements, in which each element of the set N can participate several times. In this case, no restrictions are imposed on the ratio of the values ​​of M and N, and the total number of combinations with repetitions is

An example of such a problem is the choice of M postcards from N in all possible ways.

To generate combinations with repetitions, we will use the solution to generate placement with repetitions, considered. C ++ implementation

1234567891011121314151617181920212223242526272829303132333435363738394041

#include <iostream>using namespace std;bool NextSet(int *a, int n, int m){ int j = m – 1; while (a == n && j >= 0) j–; if (j < 0) return false; if (a >= n) j–; a++; if (j == m – 1) return true; for (int k = j + 1; k < m; k++) a = a; return true;}void Print(int *a, int n) { static int num = 1; cout.width(3); cost << num ++ << “: “; for (int i = 0; i < n; i++) cost << a << ” “; cost << endl;}int main() { int n, m, *a; cost << “N = “; China >> n; cost << “M = “; China >> m; int h = n > m ? n : m; // the size of the array a is chosen as max(n,m) a = new int; for (int i = 0; i < h; i++) a = 1; Print(a, m); while (NextSet(a, n, m)) Print(a, m); cin.get(); cin.get(); return 0;}

The result of the above algorithm:

Algorithmization

## English numerology: concept and essence

The oldest knowledge, which help explain, interpret that, what angels want to tell man, called angelic numerology.

Recall, that numerology studies the vibrations of numbers, digits, their influence on people and their life processes. Another English mathematician, astrologer John Dee in the 16th century wrote a work on numbers, which he considered the universal angelic language for communicating with people.

The founder of the relatively young teaching of Angelic Numerology is Doreen Verche. Clairvoyant from America, psychologist and philosopher, author of many books. She dedicated her works to the Supreme Beings, including Gods, angels, saints. For many years she has been researching unexplained phenomena.. But it was worth it. She created the instrument, who can help you interpret the messages of your helpers from heaven. According to Doreen Verce, we regularly receive signs from angels, which warn of dangers, suggest solutions to problems in a person's life.

Angels talk to man by numbers, letters, signs. Doreen Verce's works talk about clues from invisible keepers, which are encoded in repeating digits or combinations thereof. Namely:

• on the clock;
• in the brownie, apartment, office numbering;
• in dates of birth;
• in ticket forms, like in transport, and public facilities, etc..

In all such combinations, figures hide data, who can make adjustments to the future of people. Doreen Verce has done a lot of research on the topic, dedicated to numerological calculations by date of birth. Such calculations help to identify children., who have unique abilities from birth, and not only.

Number matches, combinations we often do not notice

But it's worth paying attention to, because such a phenomenon can be an important warning. After all, angels do not send us such a sign for nothing

## Cryptographic protection of a random data generator in Python

Randomly generated numbers and data, obtained using the random module in Python, devoid of cryptographic protection. Hence, the question arises - how to achieve reliable generation of random numbers?

A cryptographically secure pseudo-random number generator is a number generator, which has features, making it suitable for use in cryptographic applications, where data security is paramount.

• All cryptographically secure generator functions return randomly generated bytes;
• The value of the random bytes, obtained as a result of using the function, depends on OS sources.
• Generation quality also depends on random sources of OS.

To ensure the cryptographic reliability of random number generation, the following approaches can be used:

• Using the secrets module to protect random data;
• Using from the os module ;
• Using a class .

An example of cryptographically secure data generation in Python:

Python

import random
import secrets

number = random.SystemRandom().random()
print(“Reliable number “, number)

print(“Reliable token bytes”, secrets.token_bytes(16))

 1 2 3 4 5 6 7 8 importrandom importsecrets number=random.SystemRandom().random() print(“Reliable number “,number) print(“Reliable token bytes”,secrets.token_bytes(16))

Output:

Shell

Reliable number 0.11139538267693572

Reliable byte token b ’ xae xa0 x91 *. Xb6 xa1 x05 = xf7 +>\r;Y xc3′

 1 2 3 Reliable number 0.11139538267693572 ReliableTokenbytesb ’ xae xa0 x91 *. Xb6 xa1 x05 = xf7 +>\r;Y xc3′

Unacceptable combinations:

• large monetary losses, tantamount to easy death will bring "2" and "4";
• scare off money - "5" and "8";
• threaten with fire "2" and "7", "9" and "5";
• business can go bust - "6" and "4";

expectations about the profitability of the event will not come true - "5" and "6".

The ability to combine numbers correctly can bring real luck and happiness into your life.. But the versatility of Eastern teaching requires a lot of attention when interpreting certain number combinations.. Though, by and large, luck in life is in the hands of the person himself and directly depends on his perseverance on the way to the goal. Go forward, do not stop there, and feng shui to help you.

## Meaning of numbers

Here is the real picture:

### 1

She is loved and revered, considering successful. Many are sure: it is this figure that is the primary source of everything that exists on earth. "One" symbolizes creative energy, unity, crystal reputation and immaculate name.

The sound of the word "one" is similar to win and honor. Combined with other numbers, it greatly enhances and complements them in a beneficial way..

### 2

Sounds sweet and easy. It is in the "two" that the harmony of Yin-Yang energy, so necessary for us, is contained, serving to create new life.

### 3

This turbulent flow of life, unprecedented development. She is in tune with growth. Which means, “Troika” carries special favorable energy in combinations with different numbers.

For instance, «3, 2, 8»Similar to the ease and accelerated development of business affairs.

### 4

we, without hesitation, love the "four", but the Cantonese dialect gives this figure consonance with death. For this reason, China bypasses it and avoids combining "4" with other numbers.. The combination of "2" and "4" speaks of an easy death, and "6" and "4" - about financial losses.

### 5

Sounds like "no" or "nothing". It is worth giving up digital combinations with the "five". Otherwise, she will make good things worthless, what other figures carry.

For instance, "5" and "6" are interpreted, as a lack of profitability.

### 6

In feng shui, it sounds similar to income, wealth.

Most consider the "six" positive, promising material well-being.

### 8

Cantonese pronounce it, denoting jewelry. This gives the G8 special luck. She is like a magnet, closely intertwined with wealth, prosperity, well-being.

Beginning with 2004 and to 2024 of the year, the Chinese calendar speaks of the increased influence of the "eight", became a lucky star for planet Earth.

### 9

In feng shui, this figure is considered very successful., denoting the complete unity of heaven and earth.

The masters of this oriental teaching call it no less positive, than the eight filled with happiness. Besides, "Nine" certainly brings physical and spiritual health, which means, and longevity.

Opinions are divided. Some draw a parallel with emptiness, and consider "zero" a bad number. But most associate it with balance and stability..

## Lucky numbers and their combinations

To summarize the above:

• there is nothing more favorable than "eight" and "nine";
• bring luck "one", "Six", "Seven";
• "Couple", sitting in front of a good number, promises good luck in business.

Using symbolic meanings of numbers, easy to find a favorable numbering for the car, choose apartment and phone number:

• any combination is favorable with "1", «6», «8»;
• win-win set, meaning constant growth of wealth, - "6" and "8";
• easy money promises "2" and "8";
• quickness and ease of receiving funds - "2", «8», «8»;
• the inevitability of getting rich profits - "7" and "8";
• hard work will receive a decent reward - "4" and "8";
• any options are successful for the numbers "7", «8», «9».

## Random number generator for lottery

You want to get random numbers without repeating. Also you don't need some numbers. Because you don't think they will fall out.. You can easily configure the mode of our number generator you need. And it will only give you useful combinations of numbers.. You don't need many different generators anymore. This RNG is versatile. This generator is easily customized for you. The generator has no restrictions on the number and range of numbers. This generation is done on the server side, not your browser. We have eliminated all factors that could affect the result of a random selection..

### New RNG generator

Our random value generator shuffles the numbers several times. We don't just generate random numbers. We first shuffle all the numbers among which we need to choose. This is done several times.. And only after that we again randomly choose a given number of numbers. This approach to random number generation guarantees a random choice.

## Permutations of n elements

Definition 3. Permutation
of n elements
any ordered set is called
these elements.

Example 7a. All kinds of permutations
multitudes, consisting of three elements {1, 2, 3} are: (1, 2, 3), (1, 3,
2), (2, 3, 1), (2, 1, 3), (3, 2, 1), (3, 1, 2).

The number of distinct permutations of n elements is denoted by Pn and
is calculated by the formula Pn=n!.

Example 8. In How Many Ways Seven Books
different authors can be arranged in one row on the shelf?Decision:this problem about the number
rearranging seven different books. There is P7=7!=1*2*3*4*5*6*7=5040
ways to arrange books.

Discussion. We see,
that the number of possible combinations can be calculated according to different rules
(permutations, combinations, placement) and the result will be different,
because. the principle of counting and the formulas themselves are different. Looking closely at
definitions, can be seen, that the result depends on several factors
at the same time.
Firstly, from that, from how many elements can we combine them
sets (how big is the total population of elements).
Secondly, the result depends on, how large are the sets of elements to us
need

And the last, It is important to know, is for us
essential order of items in a set. Let us explain the last factor on
following example

Example 9. At the parent meeting
present 20 person. How many different composition options are there
parent committee, if it should enter 5 person?Decision: In this example, we
not interested in the order of the names in the committee list. If, as a result, in his
the composition will be the same people, then the meaning for us is the same
option. Therefore, we can use the formula to calculate the number of combinations from 20 elements by 5.
Things will be different, if each committee member is initially responsible for
definite direction of work. Then with the same payroll
committee, inside it is possible 5! permutation options, which matter. amount
different (and composition, and by area of ​​responsibility) options defined in
in this case, the number of placements
of 20 elements by 5.

1. How many three-digit even numbers can be made from digits 0, 1, 2, 3, 4, 5,
6, if numbers can repeat?
Since. an even number in third place can be 0, 2, 4, 6, ie. four digits. Any of the seven digits can be in second place. Any of seven digits other than zero can be in the first place, ie. 6 opportunities. Result = 4 * 7 * 6 = 168.
2. How many five-digit numbers are there, which read the same on the left
right and right to left?
Any number can be in the first place except 0, ie. 9 opportunities. Any number can be in second place, ie. 10 opportunities. Any number from, ie. 10 opportunities. The fourth and fifth digits are predefined, they match the first and second, hence, the number of such numbers is 9 * 10 * 10 = 900.
3. There are ten subjects and five lessons per day in the class. In many ways you can
make a schedule for one day?
4. How many ways you can choose 4 delegate to the conference, if in a group
20 person?

n = C204 = (20!)/(4!*(20-4)!)=(16!*17*18*19*20)/((1*2*3*4)*(16!))=(17*18*19*20)/(1*2*3*4)=4845.

5. How many ways can eight different letters be decomposed into eight
various envelopes, if only one letter is put in each envelope?
You can put in the first envelope 1 of eight letters, in the second one of the seven remaining, in the third one of six etc.. n = 8! = 1*2*3*4*5*6*7*8 = 40320.
6. Three mathematicians and ten economists must be composed of a commission,
consisting of two mathematicians and six economists. How many ways is it
can be done?

## Lucky numbers feng shui - feng shui

The importance of numbers in feng shui is of great importance, and the Chinese, from time immemorial idolizing numerology, ready to do anything, if only car numbers, mobile phones, faxes ended in lucky numbers.

### The Meaning of Numbers in Feng Shui

The number eight in feng shui is considered extremely auspicious - this is the number of prosperity, which brings prosperity (in the language of the people of southern China, it sounds like a jewel).

The number nine symbolizes the fullness of heaven and earth and is therefore considered the most successful of all numbers.. It represents endless luck and future prosperity..

Number 9 never changes - something multiplied by 9, always leads to nine, eg: 9x3 = 27 (2+7= 9); 9x7 = 63 (6+3=9) etc.

Seven was also considered a lucky number before 2004 of the year (this was the seventh twenty-year Period), but now, since we live in the eighth Period (2004 - 2024 ) the luckier number is eight.

Numbers are considered successful combinations. 7,8 and 9 in any configuration. Chinese merchants put their price tags like this, that they end with these numbers.

for example: \$ 388, or \$27,88. or \$ 3.989.

Is considered, that such a combination of numbers brings good luck to both the seller and the buyer.

The unlucky number is currently considered a four and all, which ends with this figure. Is considered, that the four brings losses and problems.

Therefore, in China, the number 14 are credited with the same negative properties, as a number 13 in Europe.

But here's a combination - 44, 48, considered very favorable, since the sum gives 4 + 4 = eight, and among 48 contains as many as six eights .

Feng Shui numbers 2 and 3 together are considered a bad combination, as they lead to misunderstanding, can cause problems in life. But the Chinese are even more afraid of numbers. 5, considered , that it brings big problems. This is the Chinese logic ...

Numbers are considered lucky, which end in 888; 999; any combination of numbers 1, 6, 8; then, which adds up to eight when all the digits are added - 224 (2+2+4 ); 233 etc.

Favorable numbers include, which end in numbers: 1, 6, 7, 8 and 9.

For the front door, the number is considered lucky, which contains at least two digits of your date of birth. for example, person's year of birth 1972, apartment number - 72 etc.

• For financial luck, you can choose a bill with a number, ending in three eights or nines, and carry them in your wallet as a talisman, attracting money.
• Some numerologists advise you to look closely at the numbers., to those around you, since this is how the universe sends you certain signs.
• For instance, if you are going to an important meeting and there are many cars with numbers 666 ( the most destructive combination), you should be extremely careful.
• And vice versa, if there are cars with numbers, where there are many eights or nines, then the meeting will be the best way for you.
• Also, each person has their own Personal Luck Numbers.

subscribe to my channel, and as a gift you will receive a list of your personal luck numbers, which you can use in pin codes, when selling and buying housing, etc..

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