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In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra.In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, [1] India, [2] China, Germany, and Italy.
The problem of dividing the stakes became a major motivating example for Pascal in his Treatise on the arithmetic triangle. [4] [5] Though Pascal's derivation of this result was independent of Fermat's tabular method, it is clear that it also describes exactly the counting of different outcomes of + additional rounds that Fermat suggested.
Singmaster's conjecture is a conjecture in combinatorial number theory, named after the British mathematician David Singmaster who proposed it in 1971. It says that there is a finite upper bound on the multiplicities of entries in Pascal's triangle (other than the number 1, which appears infinitely many times).
The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. Visualisation of binomial expansion up to the 4th power In mathematics , the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem .
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
In 1665 Pascal posthumously published his results on the eponymous Pascal's triangle, an important combinatorial concept. He referred to the triangle in his work Traité du triangle arithmétique (Traits of the Arithmetic Triangle) as the "arithmetic triangle". [4] In 1662, the book La Logique ou l’Art de Penser was published anonymously in ...
Pascal's triangle, rows 0 through 7. The hockey stick identity confirms, for example: for n=6, r=2: 1+3+6+10+15=35.. In combinatorics, the hockey-stick identity, [1] Christmas stocking identity, [2] boomerang identity, Fermat's identity or Chu's Theorem, [3] states that if are integers, then
The two sets of three numbers which the Star of David theorem says have equal greatest common divisors also have equal products. [1] For example, again observing that the element 84 is surrounded by, in sequence, the elements 28, 56, 126, 210, 120, 36, and again taking alternating values, we have 28×126×120 = 2 6 ×3 3 ×5×7 2 = 56×210×36.