In Real life, we come across many situations, where we need to decide on combinations of certain things, may be or may not be with certain constraints. For example, choosing different fruits like apple, banana or grapes, to make fruit salad, or the safe key lock with a certain number code, which unlocks only with the correct digit combination, or at a restaurant choosing the appetizers, courses, deserts for a dinner party, or in how many ways can we set 9 books in a shelf and so on.

Such problems which are based on combinations and permutations are dealt in detail in the branch of Mathematics, called Combinatorics.

The study of Combinatorics roots from long times in B.C., starting from Mathematicians like Mahavira, Pingala, Plutarch, Abraham ibn Ezra, 13th Century legends like Leonardo Fibonacci, Jordanus de Nemore, to 17th/18th Century Pascal, Leibnitz, and Euler. George Polya, a Hungarian Analyst, one among the many famous Mathematicians, also worked on Combinatorics in the beginning of 19th century.

## Definition

Combinatorics or combinatorial analysis is the science of counting. It is the branch of mathematics that consists of selection, arrangement, and operation of elements within sets.

Combinatorial theory deals with arrangement in particular order, number of such orders, and design of arrangements. It is, to be more precise, a part of Finite mathematics under Discrete mathematics. Discrete mathematics, a branch of Mathematics dealing with study of fundamentally discrete objects, rather than continuous.
For Example: structures involving integers. Finite Mathematics being branch of Discrete math, only deals with the finite sets.
Permutations and combinations are major branches of Combinatorics other than Graph theory. The other topics which comes under combinatorics are Sets and Subsets properties, Binomial coefficients, Recurrence relation, Pascal’s Triangle, The Binomial Theorem, Congruences, Power series, Fibonacci numbers, Partitions, Stirling numbers, The principle of Inclusion and Exclusion, Exponential constant, Sperner’s theorem, The de Bruijn-Erdos theorem, Finite fields and projective planes, Systems of distinct representatives, Hall’s Theorem,  Bose’s Theorem, and Steiner triple systems.

## Formula

Does combinatorics deal with more formulas? The answer is No. Since there is a diversity in terms of being continuous, rather being more discrete this part of maths, can not be interpreted with formulas like in algebraic manipulations.  It completely depends on the logical analysis of chances to each and every new problem. Yet, some well known formulas which come under this subject would be nCr, nPr, Factorials, Partitions, binomial formulas and probability formulas:

nCr = $\frac{n!}{r!(n-r)!}$

nPr = $\frac{n!}{(n-r)!}$

n! = n * (n - 1) * (n - 2) * ...... * 1

Combinatorics include sophisticated methods of finding cardinality of sets. In analysis, the power series forming generating functions can be analyzed using this combinatorics techniques. Hyper geometric functions also seem to appear due to asymptotoic enumeration.

Combinatorial techniques can be helpful in deducing identities among functions, between products or infinite sums, even for example the famous Ramanujan Identities.

Combinatorics works related to study of patterns, designs, sets, subsets is counted as a non-enumerative branch. Some familiar ones are Latin squares, Fano plane, and Matroids.

Those who sovle Mathematical puzzles are mostly tackled with tough situations of counting combinations or symmetry. Usual techniques are not just enough to solve some combinatorial problems.

## Examples

Among many features of combinatorics, one is to find ways to prove something, like using either a counting argument or analytically. Among many famous puzzles under this subject, some are Suduko, Leonard Euler’s officers puzzle, The Tower of Hanoi, Kirkman’s School girls puzzle, Chess board – grain of wheat puzzle, and Magic squares.

Sudoku is a 9 x 9 grid, which is made of 3 rows and 3 columns, where numbers from 1 to 9 are filled in cells in such a way that each digit appears just once in each column or each row or in the small 3 x 3 sub-square.

Leonhard Euler’s Officers puzzle goes like this “Six different regiments have six officers, each one holding a different rank” The puzzle is to put these 36 officers in a square grid so that each row and each column contains one officer of each rank and one from each regiment. Similarly ‘Kirkman’s School girls’ Problem is from the charming puzzles in combinatorics.

The puzzle of chessboard-grains of wheat, is one where, one grain of wheat is placed on the first cell (square) of the chess board, two on the second cell, and so on in geometric sequence, where successive cell should be placed by double the grains as in the present cell. The solution is found to be 264 – 1.

Magic squares are another interesting puzzles in Math dealing with combination of numbers. They are squares 3 x 3, 4 x 4, 5 x 5, … and so on, where the numbers as to be filled in some particular pattern to bring a particular sum in rows, columns, diagonals called as Magic Sum.

## Applications of Combinatorics

Combinatorics works on studying pattern of objects, deciding largest, smallest or optimal structures. It is very much notable in areas like Algebra, Topology, Geometry, and Statistical Probability.

It shows up as major base for studies in Computer Science and to some extent in Physics also. Graph theory is one of the oldest studies which is derived from the concepts of combinatorics.

The applications of combinatorics is in game theory, telecommunication networks, poker hands, task distribution among employees, and to the daily activities like choosing the right dress for a party.

## Combinatorics Help

Combinatorics helps students to know how to deal with many Mathematics topics, like permutations, combinations, counting principle, probability, etc and builds ability in learners to understand Mathematical puzzles and find joy in solving them.