Topic: Repeated and Non-Repeated Substitutions

TASHKENT UNIVERSITY OF INFORMATION TECHNOLOGIES NAMED AFTER MUHAMMAD AL-KHORAZMI, MINISTRY OF INFORMATION TECHNOLOGY AND COMMUNICATION DEVELOPMENT OF THE REPUBLIC OF UZBEKISTAN

Topic: Repeated and Non-Repeated Substitutions

Prepared : Mahmudjonov Sardor

Plan:
1.Common understanding

2. Repeated replacements
3. Non-repeated substitutions
4. Conclusion


Common understanding
Replacements. It has elements
let's look at the collection. By arranging the elements of this collection in different order(write). structures (combinations) can be formed, e.g.
; ; All the elements of the given set are in each of these structures
They are only elements that are separated from each other
they differ in their places.
Definition 1. Combinations created using this method
each permutation of the elements of the set { }
is called
In fact, the phrase "swap" refers to the positions of the elements of the setif it means the act of changing, here it is created as a result of this actwe use as a structure. From this phrase in its original meaning we useA character that separates the elements of a permutation expression
as "," (comma) symbol was used above. But it is important
not, here the use of another sign, even of writing
For the sake of brevity, remove the delimiters between the elementscan also be left. This note is another that will be explained lateralso suitable for combinatorial structures.


Repeated replacements
Before in combinatoricsRepetition of its elements except for considered associationsother possible associations are also explored. For example. recurringpermutations, placements andgroupings.
The permutations studied earlier were structures such that they arethe elements were different from each other. Now the substitutionswe consider the case where the elements in the composition can be repeated.Of course, the exact same elements are new as a result of swapping their positionsno substitution occurs. Therefore, the elements of the composition A place where the elements can be repeated without changing the number .The number of substitutions is substitutions consisting of different elementswill be smaller than the number.
Suppose that is the same among n elements of some tuple
(exactly the same) first species, same second species, etc., same
Let be k- type elements, where none
are natural numbers where one is different from 1.

Definition 1. As far as possible the positions of these elements tuples (combinations) resulting from permutations ,permutations involving repetitive elements (briefly,called repeated substitutions).
among n elements the first type, the second type, etc., k is the number of repeated permutations with identical elements we define with .