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volume 4, issue 2, 2025
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“APPLICATIONS IN TEACHING THE FUNDAMENTALS OF COMBINATORICS IN
GENERAL SECONDARY SCHOOLS”
Salomov Abdinazar Abdivasikhovich
National Center for teaching teachers of Surkhandarya
region to new methods, head of the Department of pre-school,
primary and special education methods.
abdinazar2020@ mail.ru
ABSTRACT:
In this article discussed that, importance of interdisciplinary integration, some of
the requirements and conditions for the introduction of integrated education in the primary
grades, and described in detail the general features of the integrated course.
Keywords:
placements, enterprises
,
integration, interdisciplinary integration, educational
effectiveness, primary education, integration lessons.
ANNOTATSIYA:
Ushbu maqolada fanlararo integratsiya, boshlang‘ich sinflarda integratsion
ta’limni yo‘lga qo‘yishning ayrim talab va shartlari yoritilgan bo‘lib, integratsion darsning
umumiy xususiyatlari atroflicha bayon etiladi.
Kalit so'zlar:
O’rinlashtirishlar, kombinatorika, integratsiya, fanlararo integratsiya, ta'lim
samaradorligi, boshlang‘ich ta’lim, integratsion darslar.
Introduction.
As you know, no one can predict and say in advance the events taking place in
nature. For example, what will the weather be like in Tashkent in a week? Of course, no one can
say this for sure. But you can find out this information on the Internet. What is this information
based on? There is a branch of mathematics that deals with these issues. It is called “probability
theory”. The initial concepts of probability theory are concepts such as experience and event.
The initial concepts of probability theory are concepts such as experiment, event, elementary
event, probability. An experiment is an activity that produces an event. For example, tossing a
coin is an experiment. The results of the experiment are the events of the coin landing on the
“heads” or “tails” side.
Statistical definition of the probability of an event. Consider the following approximation. A
cube-shaped playing stone (duck) is thrown (Figure 1). As a result of this experiment, the
numbers one, two, three, four, five or six can fall out on its upper side. Each of these results is
considered random.
Let's throw a dice 100 times and observe how many times the event "the dice lands on the
number 6" occurs. Suppose that out of
100
experiments, the number 6 lands on 9 of them. In
1-picture
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these experiments, the number
9
is called the frequency of occurrence of the event.
The ratio of the frequency of occurrence of an event to the total number of experiments
conducted is called the relative frequency of this event. In our case, the relative frequency of the
event of the dice landing on the number 6 in 100 experiments conducted is equal to ─. In general,
a certain experiment is repeated over and over again under the same conditions, and each time
the occurrence of the event
A
of interest to us is recorded. Suppose that the total number of
experiments is n, and in m of them the event
A
occurred.
In it, the number m is the frequency of occurrence of event A, and the ratio is the relative
frequency of occurrence of event A.
In general, the experimental results show that for a large number of experiments conducted under
the same conditions, the relative frequency of occurrence of the observed event tends to a certain
stable value. It is this value that is taken as the probability of a random event. This definition is a
statistical definition of the probability of an event.
In our coin experiment above, the probability that the coin lands on heads is
is equal to .
Similarly, the probability that the coin lands on heads is is equal to.
If the occurrence of one of two events does not affect the other, such events are called
independent events,
Let's go back to the experiment of throwing a game stone. Let's assume that the game stone has a
regular shape and is made of a homogeneous material. Based on this, when the stone is thrown,
the chances of getting the numbers 1 to 6 are the same. This means that all six outcomes of the
experiment (the numbers 1, 2, 3, 4, 5, and 6) are equally likely. Also, the occurrence of one of
these events does not affect the other, that is, they are independent. Let us now consider the
occurrence of event B, that is, a multiple of 3, when the dice are rolled. This event occurs in only
two of the experiments: when the numbers 3 and 6 are rolled. We call these outcomes the
favorable (favorable) possibilities for the occurrence of event B. Out of all 6 possible, equally
likely outcomes, the number of favorable (favorable) possibilities for the occurrence of event B
is 2.
The ratio of the number of all equally likely outcomes to the number of favorable (favorable)
possibilities is called the probability of event B.
The classical definition of probability is the probability of a finite number of outcomes as a
result of an experiment.
e
1
, e
2
, e
3
,…, e
n
Let any of the elementary events occur, and let these
events form a complete set of equally likely events that are not mutually exclusive.
Suppose
n
ta e
1
, e
2
, e
3
,…, e
n
from elementary phenomena
m
k
k
k
e
e
e
,...,
,
2
1
m
If an experiment
has n equally likely outcomes, and m of them are opportunities that favor the occurrence of event
A, then
The ratio is called the probability of event A occurring and is written as:
�(�) =
�
�
The probability of getting a multiple of B − 3 in the dice throwing experiment above is:
� � =
�
�
=
2
6 =
1
3
is equal to .
The probability of an event is defined as the ratio of the number of outcomes that are favorable
(favorable) for the occurrence of event A to the number of all equally likely outcomes.
This is the classical definition of probability.
What does the fact that the probability of event B considered above is equal to , mean in practice?
Of course, this does not mean that when a dice is thrown 6 times, a multiple of 3 will appear
twice. It may appear once, three times, or not at all in 6 trials. However, if enough trials are
conducted, the relative frequency of occurrence of event B will differ very little from . In general,
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volume 4, issue 2, 2025
499
when enough experiments are conducted, the relative frequency of a random event approaches its
probability.
Suppose that event A is the occurrence of a number less than 7 when a die is thrown. It is known
that A is an inevitable event, since no number greater than 6 is thrown each time.
When a die is thrown, the occurrence of the numbers 1, 2, 3, 4, 5, or 6 is an equally likely
outcome, and their total number is n = 6. On the other hand, the possibilities that favor the
occurrence of event A are also the same outcomes, and their number is also m = 6.
Therefore, the probability of the event A occurring is P(A)= = = 1.
Thus, from the classical definition of probability, it follows that the probability of an inevitable
event is 1.
It can also be shown that the probability of an impossible event is 0.
For example, let B be the event that a 7 is thrown. Obviously, this event is impossible. This
means that the number of possibilities that make it possible for it to occur is m = 0.
Then P(B) = = = 0, that is, the probability of the impossible event B is 0.
Thus, if n is the number of equally likely outcomes and m is the number of possibilities that
make it possible for the event A to occur, then the probability of the event A is found by the
formula P(A) =.
It is known that m ≥ 0, n > 0, and m ≤ n. From these inequalities it follows that 0 ≤ ≤ 1, that is,
the probability of a random event is greater than or equal to 0 and less than or equal to 1:
0
≤ P
(
A
)
≤
1.
Exercise 1. A two-digit number is thought of, each of which has a different number of digits.
Find the probability that the number thought of is a two-digit number, each of which is a
different number of digits.
Solution: The numbers from 10 to 99 are two-digit. The two-digit numbers with the same
number of digits are 11, 22, 33, 44, 55, 66, 77, 88, 99, and there are 9 of them. There are 90
numbers from 10 to 99. The two-digit numbers with the same number of digits are
90 - 9=81 tа.
Demak, � � =
�
� =
1
81 bo
‘
ladi.
�����:
1
81 .
Exercise 2. 2000 lottery tickets were sold. Here, 1 ticket is guaranteed to win 100,000 soums, 4
tickets are guaranteed to win 50,000 soums, 10 tickets are guaranteed to win 20,000 soums, 165
tickets are guaranteed to win 5,000 soums, 400 tickets are guaranteed to win 1,000 soums, and
the remaining tickets are guaranteed to win no more than 10,000 soums. What is the probability
that one ticket will win at least 10,000 soums?
Solution: Here m = 1 + 4 + 10 + 20 = 35, n = 2000 because 35 tickets have prizes of more than
10,000 soums
�����, � � =
�
�
=
35
2000 = 0,0175.
�����: 0,0175.
Problem 3. Two dice are thrown. Find the probability that the sum of the numbers on the dice is
5 and the product is 4.
Solution: The probability that the sum of the numbers on the dice is 5
and the product is 4 is
m = 2 (5=4+1=1+4; 4=1∙4=4∙1). The number of all elementary
events is n = 36.
Demak,
A
hodisasining ro‘y berish ehtimolligi
� � =
�
�
=
2
36
=
1
18
ga teng bo‘ladi.
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500
�����:
1
18 .
1.
Analyzing statistical data helps students understand the relationships between data, think
independently, and make decisions. To develop mathematical literacy, students need to be taught
the following:
2.
Key elements:
3.
• Reading and understanding statistical data: Extracting information from tables, graphs,
or charts.
4.
• Calculating averages: Identifying measures such as the mean, median, and mode.
5.
• Expressing: Analyzing statistical results and relating them to real-life situations.
6.
• Identifying trends and changes: Observing changes in data and drawing conclusions.
Describing data using charts makes it easier to visually understand and compare them. This is an
important part of mathematical literacy and encourages students to understand statistical
processes.
7.
Types of charts:
8. Line graphs: Used to show how data changes over time.
o Example: Showing how temperatures change throughout the day.
9. Bar charts: Used to show comparisons between different categories.
o Example: Distribution of population by region.
10. Pie charts: Used to show proportions or percentages.
o Example: Percentages of the population by type of employment.
11. Histograms: Show the distribution of data.
o Example: Showing multiple return values for test results
.
Thus, to find the probability of
the event of interest, we need to conduct an experiment a sufficient number of times. This
method of finding the probability of an event is very inconvenient and time-consuming. At the
same time, if experiments are conducted on random events and the results of these experiments
are the same, that is, if there is reason to believe that their probability of occurrence is the same,
then the probability of occurrence of a random event can be found by reasoning without
conducting experiments.
Conclusion
: Analyzing statistical data and presenting it using diagrams teaches students critical
and analytical thinking. Through this, they develop the skills to understand, use, and draw
conclusions from data.
REFERENCES:
8.
M. A. Mirzaahmedov, A. A. Rahimqoriyev, Sh. N. Ismailov, M. A. Tokhtakhodjaeva
Mathematics 6th grade. Textbook for the 6th grade of general secondary schools. T.: Teacher,
2017.
9.
Sh. A. Alimov, O. R. Kholmukhamedov, M. A. Mirzaahmedov Algebra. Textbook for
the 7th grade of general secondary schools. T.: Teacher, 2017.
10.
U.Kh.Khonkulov, Elements of the stochastic direction of mathematics.-T.: Science and
technology”, 2017.
11.
U.Kh.Khonkulov, Combinatorics issues in teaching mathematics. Gulistan. 2008.
12.
Bunimovich E. A., Bulychev V. A. Reliability and statistics. 5-9 kl.: Posobie dlya
obshcheobrazovat. fly deluded. - M.: Drofa, 2002.
13.
Makarychev Yu. N. Algebra: elementi statistici i teorii veroyatnostey: ucheb. posobie
dlya uchashchixsya 7-9 kl. public education uchrejdenii / Yu. N. Makarychev, N. G.
Mindyuk; pod ed. S. A. Telyakovsky. - 2nd izd. - M.: Prosveshchenie, 2004.
14.
Mordkovich A. G., Semenov P. V. Sobytia. Believability. Statisticheskaya
obrabotka dannyx: Dop. Paragraph k course algebra 7-9 kl. public education Uchrejdeny. - M.:
Mnemosina, 2003.
15.
A.A.Salomov An integrated approach to in-service training proficiency factors 2022 –
y 142.p.
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volume 4, issue 2, 2025
501
16.
A.A.Salomov "Razvitie, umenie i navykov u chashchikhsya mejspredmetnoy funktsii
mathematiki" //Dissertatsionnyy issleddovanie//. Tashkent-1996
17.
A.A. Salomov "Development, art and skills and the mathematical function of the
subject" Tashkent: -1996. //Dissertatsionnyy issledovanie//.
Internet resources:
1. http:/www.markaz.tdi.uz
2. @milliymarkaz_aloqabot
3.
4. http://www.dtm.uz
5. http:/www.markaz.tdi.uz
