Combinatorics formulas. One of the concepts of combinatorics One of the concepts of combinatorics 9 letters crossword puzzle

COMBINATORICS

Combinatorics is a branch of mathematics that studies the problems of selecting and arranging elements from a certain basic set in accordance with given rules. Formulas and principles of combinatorics are used in probability theory to calculate the probability of random events and, accordingly, obtain the laws of distribution of random variables. This, in turn, allows us to study the patterns of mass random phenomena, which is very important for a correct understanding of the statistical patterns that manifest themselves in nature and technology.

Rules for addition and multiplication in combinatorics

Sum rule. If two actions A and B are mutually exclusive, and action A can be performed in m ways, and B in n ways, then one of these actions (either A or B) can be performed in n + m ways.

Example 1.

There are 16 boys and 10 girls in the class. In how many ways can you assign one duty officer?

Solution

Either a boy or a girl can be assigned to duty, i.e. the duty officer can be any of the 16 boys or any of the 10 girls.

Using the sum rule, we find that one duty officer can be assigned in 16+10=26 ways.

Product rule. Let there be k actions required to be performed sequentially. If the first action can be performed in n 1 ways, the second action in n 2 ways, the third in n 3 ways, and so on until the kth action that can be performed in n k ways, then all k actions together can be performed:

ways.

Example 2.

There are 16 boys and 10 girls in the class. In how many ways can two duty officers be appointed?

Solution

Either a boy or a girl can be appointed as the first person on duty. Because There are 16 boys and 10 girls in the class, then you can appoint the first person on duty in 16+10=26 ways.

After we have chosen the first duty officer, we can choose the second one from the remaining 25 people, i.e. 25 ways.

According to the multiplication theorem, two attendants can be selected in 26*25=650 ways.

Combinations without repetition. Combinations with repetitions

A classic problem in combinatorics is the problem of the number of combinations without repetitions, the content of which can be expressed by the question: how many ways Can choose m from n different items?

Example 3.

You must choose 4 out of 10 different books available as a gift. In how many ways can this be done?

Solution

We need to choose 4 books out of 10, and the order of choice does not matter. Thus, you need to find the number of combinations of 10 elements of 4:

.

Consider the problem of the number of combinations with repetitions: there are r identical objects of each of n different types; how many ways Can choose m() from these (n*r) items?

.

Example 4.

The pastry shop sold 4 types of cakes: Napoleons, eclairs, shortbread and puff pastries. In how many ways can you buy 7 cakes?

Solution

Because Among 7 cakes there may be cakes of the same type, then the number of ways in which 7 cakes can be bought is determined by the number of combinations with repetitions of 7 to 4.

.

Placements without repetition. Placements with repetitions

A classic problem in combinatorics is the problem of the number of placements without repetitions, the content of which can be expressed by the question: how many ways Can choose And post By m different places m from n different items?

Example 5.

Some newspaper has 12 pages. It is necessary to place four photographs on the pages of this newspaper. In how many ways can this be done if no page of the newspaper should contain more than one photograph?

Solution.

In this task, we do not just select photographs, but place them on certain pages of the newspaper, and each page of the newspaper should contain no more than one photograph. Thus, the problem is reduced to the classical problem of determining the number of placements without repetitions of 12 elements of 4 elements:

Thus, 4 photos on 12 pages can be arranged in 11,880 ways.

Also a classic problem in combinatorics is the problem of the number of placements with repetitions, the content of which can be expressed by the question: how many ways Can Youbarmy And post By m different places m from n items,Withready which There is the same?

Example 6.

The boy still had stamps with the numbers 1, 3 and 7 from his board game set. He decided to use these stamps to put five-digit numbers on all the books to create a catalogue. How many different five-digit numbers can a boy create?

Permutations without repetition. Permutations with repetitions

A classic problem in combinatorics is the problem of the number of permutations without repetition, the content of which can be expressed by the question: how many ways Can post n various items on n different places?

Example 7.

How many four-letter “words” can you make from the letters of the word “marriage”?

Solution

The general population is the 4 letters of the word “marriage” (b, p, a, k). The number of “words” is determined by the permutations of these 4 letters, i.e.

For the case when among the selected n elements there are identical ones (selection with return), the problem of the number of permutations with repetitions can be expressed by the question: In how many ways can n objects located in n different places be rearranged if among n objects there are k different types (k< n), т. е. есть одинаковые предметы.

Example 8.

How many different letter combinations can be made from the letters of the word "Mississippi"?

Solution

There is 1 letter "m", 4 letters "i", 3 letters "c" and 1 letter "p", for a total of 9 letters. Therefore, the number of permutations with repetitions is equal to

BACKGROUND SUMMARY FOR THE SECTION "COMBINATORICS"

One of the concepts of combinatorics

First letter "s"

Second letter "o"

Third letter "h"

The last letter of the letter is "e"

Answer for the question "One of the concepts of combinatorics", 9 letters:
combination

Alternative crossword questions for the word combination

mathematical term

Mathematical term

Connection, arrangement of something, forming a unity, whole

A connection that forms a unity, a whole

Definition of the word combination in dictionaries

Explanatory dictionary of the Russian language. S.I.Ozhegov, N.Yu.Shvedova. The meaning of the word in the dictionary Explanatory Dictionary of the Russian Language. S.I.Ozhegov, N.Yu.Shvedova.
-I, Wed see combine, -xia. A connection, an arrangement of something that forms a unity, a whole. S. sounds. Beautiful village colors. * In combination with someone, meaning. prepositions with those n. - together, next to someone. Talent combined with performance.

Explanatory dictionary of the Russian language. D.N. Ushakov The meaning of the word in the dictionary Explanatory Dictionary of the Russian Language. D.N. Ushakov
combinations, cf. only units Action according to verb. combine. Thanks to a skillful combination of underground work and legal work, the Bolsheviks managed to become a serious force in open workers' organizations. History of the CPSU(b). Combination of theory and practice. Marriage combination...

New explanatory dictionary of the Russian language, T. F. Efremova. The meaning of the word in the dictionary New explanatory dictionary of the Russian language, T. F. Efremova.
Wed The process of action by value. nesov. verb: combine, combine (1*). Status by value nesov. verb: combine (1*).

Encyclopedic Dictionary, 1998 The meaning of the word in the dictionary Encyclopedic Dictionary, 1998
see Combinatorics.

Wikipedia Meaning of the word in the Wikipedia dictionary
In combinatorics, a combination of n through k is a set of k elements selected from a given set containing n different elements. Sets that differ only in the order of the elements are considered the same, this is how combinations differ from placements...

Examples of the use of the word combination in literature.

The fact is that combination adventurism with acute problems, dialogism, confession, life and preaching is not at all something completely new and has never happened before.

So the master painted a wonderful self-portrait, which is now in the Frick collection in New York and amazes with its peculiar combination an ironic grin with the majesty of a solemn ceremony.

Moreover, is it not possible to assert that the modern crisis of authoritarianism is a fluctuation, a rare combination political planets, which will not happen again in the next few hundred years?

Goya, who extracted from aquatint, often in combination with etching, expressive contrasts of dark tones and sudden impacts of light spots, and the French artist L.

Nero praised Acte in elegant verses, and some of them became popular, especially two poems where he praised Acte combination child and woman, chastity and passion.