Properties About Combination

Introduction 

In mathematics, there are situations, when there are numerous ways to solve an equation and it can have multiple answers as well, referred to as combinatory. A combination in probability is how the selection of items is done from a pile of objects kept arranged or in an orderly fashion and it is done in natural numbers.  The introduction of Permutation combinations has been done through the hands of French Mathematicians. Bigger equations and problems can be easily solved through the properties of Combinations; one of the main properties of combinations is the similarity between the formula’s denominator and in that case, all the possible values of an unknown value. 

Discussion

What are the properties of Permutation?

A set of objects has varied characteristics such as colour, shape, properties and others. The way of arranging the items or objects be varied, and a person can opt for any way of arranging the items based on choice. For instance, the arrangement can be done based on colour, shape or size and therefore, the number of ways a group of items can be arranged is called its permutations. There can number of permutations for several objects, for instance, for one object, there is one way of permutation, and for 3 objects there can be 3*2= 6 ways of permutations. The properties of permutation can be described below- 

• Permutation works with unordered elements.
• It deals with the number of arrangements that can be created from a provided set of items or objects.
• Permutation symbolizes arrangement.
• Permutations are applied in places where diversity exists in the objects. 
• Permutation can be of three types one with allowed repetition, another where repetition is not allowed and the other with multiple sets.
• Factorials are used in the calculation of permutations. 

 A real-life example can be for instance a class that has been provided with 5 different types of chocolates, with five different flavours as cocoa, mint, butterscotch so on. There are 15 students in the class, therefore,

r = 15 children

N (objects) = 5 types of toffee

Thus, the number of permutations possible with repetitions can be 5 to the power 15 = 30, 51,578,125 permutations. 

Use of probability and statistic 

“Combinatorics” is a part of probability statistics that deals with counting objects and ordering them accordingly. Probability counting is important in daily life, and that is why it is applied in various applied statistics. Rolling of dice, putting locker combinations, solving military cases, Bank Pins are examples. Sometimes, in daily lives, the permutation and combination do not work together, as the order does not matter always, however, in certain cases; both permutation and combination add up to give results to equations. Permutations and combinations are a part of probability theory. Apart from this, combinatorics in probability has applications in the mathematics field, such as geometrical applications where graphs are assessed based on the number of possible diagonals in a pentagon or hexagon, the number of possible straight lines that can be passed through points. 

• Probability can be used in calculating official seating arrangements, arranging or verifying codes of government level importance. 
• The medical field uses the occurrence capacity of any disease through permutation and combinations. 
• Combinations are applicable in network security systems, banks, cryptography, coding and decoding, pattern analysis and others.
• In research fields of applied physics, combinations can be applicable in string theory, gases combination, also in descriptive research. 

Difference between permutation and combination 

In permutation, the order of choosing items matters because of its representation, while in combination, the order in which matters or items are chosen among n number of items does not matter. For the working of combination, a “k set of objects” is chosen from an “n set of objects” for the production of subsets that are in order. Permutation and combination initially differ in formulas and usages; also, their process of calculation differs as well. The calculation of combination can be explained as initially a set of objects grouped in possible groupings of three objects such as colours. Then, the previous example can be explained as only one combination can be made from the toffee permutations of number 2*3= 6 permutations. 

Properties about combinations are listed below- 

• Combinations area derived from permutations when r objects are already signified.
• The possible number of ways of selecting “n” objects out of n objects is always a one.
• The combination is usually done from pre-derived groupings. 
• The possible number of selection of one object is (n-1) = n items. 
• Repetition is not allowed in combination.

Conclusion 

In summarization, it can be said that permutations and combinations are a part of the probability theory of statistics and an integral part of the counting system called combinatorics. The use of permutation combination in daily life is wide, such as decision making related to official documents, in geometry, string theory and others. Although permutation and combination differ from each other in terms of their calculation, both of them a complimentary to each other, for instance, for calculating combination, permutations are required. The combination plays an immense role in solving bigger equations through research analysis, for instance, in scientific researches of subjects such as psychology, mathematics, and physics.