CHAPTER I
INTRODUCTIONS
Background of the Study
We often face the problems associated with order, arrangement or the like. In such cases there are two types of arrangement that is arrangement with respect to the order and arrangement which does not pay attention to the order. In the case of mathematical structure which respect of the order called a permutation, while those not referred of the order called by a combination.
This paper will explain about the use of permutations in daily life. As we know, there are many benefits of permutations in daily life. For the example, permutations is applicable in problem drafting committee consisting of a chairman, secretary and treasurer of the order in which the consideration. Unconscious, when we play a games using dice, coin toss, bridge card games (rummy) or the like that, we use the concept of probability. And permutation is a part of probability that we have studied since in Senior High School until now. Permutation is an arrangement of distinct or different sequences formed by partial or whole objects or elements taken from a group of objects or elements available. The composition of the permutations of the order means AB with BA is calculated differently.
The Problems of the Study
What is definition of permutations?
What are the benefits of permutations?
How does permutations use in daily life?
The Purposes of the Study
To know about definition of permutations
To know about the benefit of permutations
To know how permutations use in daily life
Key Term
Permutations are an arrangement that can be formed from a collection of objects taken partly or wholly by considering the sequence.
CHAPTER II
DISCUSSIONS
Definitions of Permutations
Calculation of material opportunities that often popularized by the term probability first introduced by Blaise Pascal and Pierre de Fermat in the 17th century through the game of dice. This is the end of game developing game dice games such as coin toss, bridge card games (rummy) and other games. Therefore, the concept was born through a game of chance. During its development, the calculation of the chances of getting serious attention from scientists because it has a very important role in the development of other sciences, such as physics modern science, Statistics, and others.
At first opportunity only made in gambling games. A gambler wants a big win, so enlist the help of a mathematician to win the game maneuver. But due to the rapid development of the theory of chances, eventually used in the fields of politics, economics, weather forecasting and scientific research. Opportunity theory relates to the calculation of odds or likelihood of an event. An event is part of a larger event or sample space. To determine the odds of an event need to determine in advance how many events it can happen and how much space of the sample can occur.
In this moment, opportunity theory developed to be probability theory which both of them discuss about arrangement, order, trial, sample, event and etc. In mathematics, probability is a common sense expanded by calculations and this theory has classified into two types of arrangement which has contra definitions for one each other. For the first is arrangement which respect to the order that called Permutations and the second is arrangement which does not pay attention to the order that called combination.
Permutation is an arrangement of distinct or different sequences formed by partial or whole objects or elements taken from a group of objects or elements available. The composition of the permutations of the order means AB with BA is calculated differently. Example for the arrangement which respect of the order is if there are 3 students A, B, C will be selected as the Chairman, Secretary and Treasurer with each student may only be selected for one position only, then we can using the multiplication principle to determine the number of arrangements possible committee, namely :
- to determine chairman may be made in 3 ways,
- after determined chairman, secretary may be determine in 2 ways,
- and also after determined chairman and secretary, treasurer may be determine in 1 way.
So, amount arrangement that perhaps the committee structure are 3 x 2 x 1 = 6 ways. And the conclusion for the number of selections that may be happen there are 6 results, they are : (A,B,C), (B,C,A), (C,A,B), (C,B,A), (A,C,B) and (B,A,C). This is because, for the selection results (A, B, C) means that A be the Chairman, Secretary B, and C be the Treasurer, it will be different with the selection results (B, C, A) which means as Chairman B, C as A Secretary and a Treasurer.
There are three kinds of permutations, namely:
1. Cyclic Permutations
Cyclical permutations for n = the number of ways a circle object with different order. The number of cyclic permutations of n elements is:
Example:
How many ways there are 7 people sitting around the table can occupy seven seats with a different order?
Answer:
Many ways sitting there (7-1)! = 6! = 6. 5. 4. 3. 2. 1 = 720 ways.
2. Permutation Which Contain Same Common Elements
Suppose it is evident that there are k n elements and each element appears q1, q2, q3... , qn times. Permutations of n elements are:
Example:
Specify the number of permutations of the letters contained in the student OSIS! Permutations of n unsure are:
Answer:
O = 1
S = 2
I = 1
n = number of the letters = 4, Thus
3. Permutation Which Contain Different Elements
Suppose n different elements are known. The number of permutations of r elements (r less than or equal to n) of n elements taken are:
Example:
Find 2 specify the many permutations of the elements of the letters A, B, C, D!
Answer:
The Benefits of Permutations
- To count the number of ways for determine arrangement
In our daily life, we always count things. For the example when we are dressing. We may count the number of possible ways to choose a pair of trousers, a shirt and a jacket from the wardrobe for a proper match. Sometimes, the number we count is so huge that the counting job is not easy. However, some kinds of mathematics may help us do it in a systematic way.
Oftentimes, we unconscious that the benefit of permutations is happen in our daily life. And permutations is be able to count the number of ways for determine something which respect with the order. For the example to determine election of structure organization. We are given the option to specify the position of the post of chairman, assistant of chairman, secretary, treasurer, etc.. Moreover, the permutation can also be used to determine the number of ways in the preparation of a series of letters in the word or the other.
- Be distinguishing for one arrangement to another arrangement.
By using permutations, we can find the number of ways classification for each arrangement. For the example in the election of structure organization, we can get result of chairman, assistant of chairman, secretary, treasurer, etc. Which all of them are different one each other. They have different position and job that we can make it same. So benefit of permutation is to distinguishing for one arrangement to another arrangement.
The Use of Permutations in Daily Life
A. Election of Organization’s Structure
From our study in the elementary school until in this university, we always find the election organization’s structure. And it also be pre-requirement in each community around us. Structure of organization is very important for us. It can manage the member of class or community and facilitate them.
For the example, if there are 3 students A, B, C will be selected as the Chairman, Secretary and Treasurer with each student may only be selected for one position only, then we can using the multiplication principle to determine the number of arrangements possible committee. The ways of election has explained before.
B. To Determine Number of the Sequence of Letters and Numbers
A father who was waiting for his wife in labor, he has prepared ANI name for the baby that will be born. Moments later, the midwife told me that she would give birth to twins the father thought, how many twins that can be named using the letters in the word ANI?
From this story, the old man took to the first letter of the three available letters. Next take a letter to the second letter of the two remaining letters. Finally, take a single letter for the rest of the third letter.
If we find the number of letter ANI to be another name by manual that can be:
- ANI
- INA
- AIN
- IAN
- NIA
- NAI
Based on the basic rules of counting, the number of arrangements 3 elements ( letters ) differ in no particular order with no elements ( letters ) that may be repeated is = 3 x 2 x 1 = 6 arrangement .
In general, the preparation of n distinct elements in a specific order with no repeated elements is called a permutation of n elements.
C. Determine the amount of variance number for SIM Card
Prime telephone number used by the operator to have a lot of variance, it is because the number operator scrambles to produce different numbers. In general, the number of digits used operator is twelve, but of the 12 digit number in front of him is the identity of the operator. For example IM3, in front use the number 085 and the dot dot dot. From these numbers the rest scrambled by the operator, so many kinds of numbers that can be provided by mobile operators as much as 1,000,000,000 IM3 is number variants. This can be obtained by the following calculation,
First we create 12 columns, then proceed with the possibilities of each digit column because the front should wear digit 085. For the first then the second and the third column only have 1 option. Due to the next column and there is no requirement that the number of 10 digits wide then there are 10 numbers per column option. After that, multiply every column.
Then obtained 1X1X1X10X10X10X10X10X10X10X10X10 = 1,000,000,000 variance number.
| 1 | 1 | 1 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 |
D. To determine the arrangement of colors on the flag
As we know, the color of Indonesian’s flag is red and white, whereas if position of both colors (red and white) is altered, then the meaning of the different colors and the flag will be turned into a Polish state flag.
Things that seem trivial permutations proved using concepts that are applied in daily life.
E. Application of the science of encryption or security code (password)
In this modern era, we often face with password. Password used by the user so that it becomes something that belong more privacy. For the example in mobile phone or netbook, both are familiar with his name password. Usually someone makes a difficult password or the easy one. Difficult or not the password was influenced by the choice of the user to create a password. The more digits are used increasingly difficult eating hackers to break into password, it is called because the more digits, the more possible arrangements of digits required, if the password consists of a 2-digit number then obtained 100 variance, it is necessary try to break into the possibility of 100 Next time try. And so on.
CHAPTER III
SUMMARY OF THE POINTS
Ø Permutations is an arrangement of distinct or different sequences formed by partial or whole objects or elements taken from a group of objects or elements available which respect of the order.
Ø There are three kinds of permutations, namely: Cyclic Permutations, Permutation Which Contain Same Common Elements and Permutation Which Contain Different Elements.
Ø Spread all over, the benefit of permutations is to count the number of ways for determine arrangement which respect with the order. Permutations also can be distinguishing for one arrangement to another arrangement.
Ø The use of permutations in daily life is very much. Actually we unconscious that we have done some activity by applying for permutations. The example of use permutations in daily life are :
- In Election of Organization’s Structure
- To determine number of the sequence of letters and numbers
- To determine amount of variance number for SIM Card
- To determine the arrangement of colors on the flag
- Application of the science of encryption or security code (password)
REFERENCE
Munthe, Joseph Andreas. 2012. Pengertian Permutasi. (Online), (http://josephmunthe.blogspot.com/2012/07/sejarah-peluang.html), retrification at 21 October 2013
Turmudi & Harini, Sri. 2008. Metode Statistika: Pendekatan Teoretis dan Aplikatif. Malang: UIN Malang Press.
Valiant, Vip. 2013. Permutasi, Kombinasi, dan Peluang. (Online), (http://vipvaliant26.blogspot.com/2013/07/makalah-permutasi-kombinasi-dan-peluang.html). retrification at 21 October 2013
Wikipedia. 2013. Permutasi. (Online), (http://id.wikipedia.org/wiki/Permutasi), retrification at 21 October 2013
http : // users.math.yale.edu/lecturer13.pdf-21 oktober 2013, retrification at 22 October 2013
yoyok1985.files.wordpress.com/2009/01/permutasi.pdf, retrification at 22 October 2013
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