Combinatorics is the branch of mathematics which is related to counting.

**Introduction**

Combinatorics is the branch of
mathematics which is related to counting. It deals with arrangements of objects
as well as enumeration, that is, counting of objects with specific properties.
The roots of the subject can be traced as far back as 2800 BCE when it was used
to study magic squares and patterns within them.

English physicist and
mathematician Sir Isaac Newton, most famous for his law of gravitation, was
instrumental in the scientific revolution of the 17th century. Newton’s belief
in the “Persistance of patterns” led to his first significant mathematical
discovery, the generalization of the expansion of binomial expressions.

Newton discovered Binomial
Theorem which he claimed the easiest way to solve the quadratures of curves.
This discovery is essential in understanding probability. The generalized
version of the Binomial Theorem, the Multinomial Theorem, applies to multiple
variables. It is widely used in Combinatorics and Statistics.

He was the first to use
fractional indices and to employ coordinate
geometry to derive
solutions to Diophantine
equations. He approximated
partial sums of the harmonic series by logarithms (a precursor to
Euler’s summation formula)
and was the
first to use power series with confidence and to
revert power series. Newton’s Newton (1643*−*1727) work on infinite series was
inspired by Simon Stevin’s decimals.

In 1705, he was knighted by Queen
Anne of England, making him Sir Isaac Newton. Newton made discoveries in optics
and theory of motions. Along with mathematician Leibnitz, Newton is credited
for developing essential theories of calculus.

Combinatorics has many real life
applications where counting of objects are involved. For example, we may be
interested to know if there are enough mobile numbers to meet the demand or the
number of allowable passwords in a computer system. It also deals with counting
techniques and with optimisation methods, that is, methods related to finding
the best possible solution among several possibilities in a real problem. In
this chapter we shall study counting problems in terms of ordered or unordered
arrangements of objects. These arrangements are referred to as permutations and
combinations. Combinatorics are largely used in the counting problems of
Network communications, Cryptography, Network Security and Probability theory.
We shall explore their properties and apply them to counting problems.

Consider another situation: We all know that our electricity consumer card number is of the form B : C, where A denotes the electrical substation /larger capacity transformer number, B denotes the smaller capacity electricity transformer number and C denotes the consumer number. There may be conditions that to each substation certain maximal number of transformer can only be linked and with a particular transformer certain maximal number of consumer connection can only be linked. Now the question of deciding, whether a new Transformer/Substation needs to be erected, can be made by the count of the number of consumer connections linked with a substation transformer. How to get that count? This count can be easily arrived by the use of counting principles.

Tags : Mathematics , 11th Mathematics : UNIT 4 : Combinatorics and Mathematical Induction

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11th Mathematics : UNIT 4 : Combinatorics and Mathematical Induction : Combinatorics and Mathematical Induction: Introduction | Mathematics

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