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In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial …

1000th row of Pascal's triangle, arranged vertically, with grey-scale representations of decimal digits of the coefficients, right-aligned. The left boundary of the image corresponds roughly to the graph of the logarithm of the binomial coefficients, and illustrates that they form a log-concave sequence.

Image: Pascal triangle small

The binomial coefficients can be arranged to form Pascal's triangle.

Visualisation of binomial expansion up to the 4th power

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1. Binomial coefficient – In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly, a coefficient is indexed by a pair of integers n ≥ k ≥0 and is written. It is the coefficient of the xk term in the expansion of the binomial power n. The value of the coefficient is given by the expression n. k, arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascals triangle. The properties of binomial coefficients have led to extending the definition to beyond the case of integers n ≥ k ≥0. Andreas von Ettingshausen introduced the notation in 1826, although the numbers were known centuries earlier, the earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, by Halayudha, on an ancient Sanskrit text, Pingalas Chandaḥśāstra. In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī, alternative notations include C, nCk, nCk, Ckn, Cnk, and Cn, k in all of which the C stands for combinations or choices. Many calculators use variants of the C notation because they can represent it on a single-line display, in this form the binomial coefficients are easily compared to k-permutations of n, written as P, etc. For natural numbers n and k, the binomial coefficient can be defined as the coefficient of the monomial Xk in the expansion of n, the same coefficient also occurs in the binomial formula, which explains the name binomial coefficient. This shows in particular that is a number for any natural numbers n and k. Most of these interpretations are easily seen to be equivalent to counting k-combinations, several methods exist to compute the value of without actually expanding a binomial power or counting k-combinations. It also follows from tracing the contributions to Xk in n−1, as there is zero Xn+1 or X−1 in n, one might extend the definition beyond the above boundaries to include =0 when either k > n or k <0. This recursive formula then allows the construction of Pascals triangle, surrounded by white spaces where the zeros, or the trivial coefficients, a more efficient method to compute individual binomial coefficients is given by the formula = n k _ k. = n ⋯ k ⋯1 = ∏ i =1 k n +1 − i i and this formula is easiest to understand for the combinatorial interpretation of binomial coefficients. The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, the denominator counts the number of distinct sequences that define the same k-combination when order is disregarded. Due to the symmetry of the binomial coefficient with regard to k and n−k, calculation may be optimised by setting the limit of the product above to the smaller of k. This formula follows from the formula above by multiplying numerator and denominator by. As a consequence it involves many factors common to numerator and denominator and it is less practical for explicit computation unless common factors are first cancelled

2. Mathematics – Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

3. Recursion – Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic, the most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines a number of instances, it is often done in such a way that no loop or infinite chain of references can occur. The ancestors of ones ancestors are also ones ancestors, the Fibonacci sequence is a classic example of recursion, Fib =0 as base case 1, Fib =1 as base case 2, For all integers n >1, Fib, = Fib + Fib. Many mathematical axioms are based upon recursive rules, for example, the formal definition of the natural numbers by the Peano axioms can be described as,0 is a natural number, and each natural number has a successor, which is also a natural number. By this base case and recursive rule, one can generate the set of all natural numbers, recursively defined mathematical objects include functions, sets, and especially fractals. There are various more tongue-in-cheek definitions of recursion, see recursive humor, Recursion is the process a procedure goes through when one of the steps of the procedure involves invoking the procedure itself. A procedure that goes through recursion is said to be recursive, to understand recursion, one must recognize the distinction between a procedure and the running of a procedure. A procedure is a set of steps based on a set of rules, the running of a procedure involves actually following the rules and performing the steps. An analogy, a procedure is like a recipe, running a procedure is like actually preparing the meal. Recursion is related to, but not the same as, a reference within the specification of a procedure to the execution of some other procedure. For instance, a recipe might refer to cooking vegetables, which is another procedure that in turn requires heating water, for this reason recursive definitions are very rare in everyday situations. An example could be the procedure to find a way through a maze. Proceed forward until reaching either an exit or a branching point, If the point reached is an exit, terminate. Otherwise try each branch in turn, using the procedure recursively, if every trial fails by reaching only dead ends, return on the path led to this branching point. Whether this actually defines a terminating procedure depends on the nature of the maze, in any case, executing the procedure requires carefully recording all currently explored branching points, and which of their branches have already been exhaustively tried. This can be understood in terms of a definition of a syntactic category. A sentence can have a structure in which what follows the verb is another sentence, Dorothy thinks witches are dangerous, so a sentence can be defined recursively as something with a structure that includes a noun phrase, a verb, and optionally another sentence

4. Sanskrit – Sanskrit is the primary liturgical language of Hinduism, a philosophical language of Hinduism, Buddhism, and Jainism, and a literary language and lingua franca of ancient and medieval South Asia. As a result of transmission of Hindu and Buddhist culture to Southeast Asia and parts of Central Asia, as one of the oldest Indo-European languages for which substantial written documentation exists, Sanskrit holds a prominent position in Indo-European studies. The body of Sanskrit literature encompasses a rich tradition of poetry and drama as well as scientific, technical, philosophical, the compositions of Sanskrit were orally transmitted for much of its early history by methods of memorization of exceptional complexity, rigor, and fidelity. Thereafter, variants and derivatives of the Brahmi script came to be used, Sanskrit is today one of the 22 languages listed in the Eighth Schedule of the Constitution of India, which mandates the Indian government to develop the language. It continues to be used as a ceremonial language in Hindu religious rituals and Buddhist practice in the form of hymns. The Sanskrit verbal adjective sáṃskṛta- may be translated as refined, elaborated, as a term for refined or elaborated speech, the adjective appears only in Epic and Classical Sanskrit in the Manusmṛti and the Mahabharata. The pre-Classical form of Sanskrit is known as Vedic Sanskrit, with the language of the Rigveda being the oldest and most archaic stage preserved, Classical Sanskrit is the standard register as laid out in the grammar of Pāṇini, around the fourth century BCE. Sanskrit, as defined by Pāṇini, evolved out of the earlier Vedic form, the present form of Vedic Sanskrit can be traced back to as early as the second millennium BCE. Scholars often distinguish Vedic Sanskrit and Classical or Pāṇinian Sanskrit as separate dialects, although they are quite similar, they differ in a number of essential points of phonology, vocabulary, grammar and syntax. Vedic Sanskrit is the language of the Vedas, a collection of hymns, incantations and theological and religio-philosophical discussions in the Brahmanas. Modern linguists consider the metrical hymns of the Rigveda Samhita to be the earliest, for nearly 2000 years, Sanskrit was the language of a cultural order that exerted influence across South Asia, Inner Asia, Southeast Asia, and to a certain extent East Asia. A significant form of post-Vedic Sanskrit is found in the Sanskrit of Indian epic poetry—the Ramayana, the deviations from Pāṇini in the epics are generally considered to be on account of interference from Prakrits, or innovations, and not because they are pre-Paninian. Traditional Sanskrit scholars call such deviations ārṣa, meaning of the ṛṣis, in some contexts, there are also more prakritisms than in Classical Sanskrit proper. There were four principal dialects of classical Sanskrit, paścimottarī, madhyadeśī, pūrvi, the predecessors of the first three dialects are attested in Vedic Brāhmaṇas, of which the first one was regarded as the purest. In the 2001 Census of India,14,035 Indians reported Sanskrit to be their first language, in India, Sanskrit is among the 14 original languages of the Eighth Schedule to the Constitution. The state of Uttarakhand in India has ruled Sanskrit as its official language. In October 2012 social activist Hemant Goswami filed a petition in the Punjab. More than 3,000 Sanskrit works have been composed since Indias independence in 1947, much of this work has been judged of high quality, in comparison to both classical Sanskrit literature and modern literature in other Indian languages