The triangle demonstrates many mathematical properties in addition to showing binomial coefficients.

His Traité du triangle arithmétique (" Treatise on the Arithmetical Triangle ") of 1653 described a convenient tabular presentation for binomial coefficients, now called Pascal's triangle.

where are the binomial coefficients.

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

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem.

Arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascal's triangle.

The properties of binomial coefficients have led to extending the meaning of the symbol beyond the basic case where n and k are nonnegative integers with ; such expressions are then still called binomial coefficients.

The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, due to Halayudha, on an ancient Hindu classic, Pingala's chandaḥśāstra.

In about 1150, the Hindu mathematician Bhaskaracharya gave a very clear exposition of binomial coefficients in his book Lilavati.

This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients ( counting problems for which the answer is given by a binomial coefficient expression ), for instance the number of words formed of n bits ( digits 0 or 1 ) whose sum is k is given by, while the number of ways to write where every a < sub > i </ sub > is a nonnegative integer is given by.

One has a recursive formula for binomial coefficients

This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century, but they were known to many mathematicians who preceded him.

The final expression follows from the previous one by the symmetry of x and y in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical.

The binomial coefficients 1, 2, 1 appearing in this expansion correspond to the third row of Pascal's triangle.

# the nth row of the Pascal's Triangle will be the coefficients of the expanded binomial.

The coefficients that appear in the binomial expansion are called binomial coefficients.

The sum is taken over all combinations of nonnegative integer indices k < sub > 1 </ sub > through k < sub > m </ sub > such that the sum of all k < sub > i </ sub > is n. That is, for each term in the expansion, the exponents of the x < sub > i </ sub > must add up to n. Also, as with the binomial theorem, quantities of the form x < sup > 0 </ sup > that appear are taken to equal 1 ( even when x equals zero ).

Just like " n choose k " are the coefficients when you raise a binomial to the n < sup > th </ sup > power ( e. g. the coefficients are 1, 3, 3, 1 for ( a + b )< sup > 3 </ sup >, where n = 3 ), the multinomial coefficients appear when one raises a multinomial to the n < sup > th </ sup > power ( e. g. ( a + b + c )< sup > 3 </ sup >)

The inverse of the Hilbert matrix can be expressed in closed form using binomial coefficients ; its entries are

A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.

The arithmetical triangle — a graphical diagram showing relationships among the binomial coefficients — was presented by mathematicians in treatises dating as far back as the 10th century, and would eventually become known as Pascal's triangle.

An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle.

* Pascal's triangle, a geometric arrangement of the binomial coefficients in a triangle

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients.

This construction is related to the binomial coefficients by Pascal's rule, which says that if

Pascal's triangle determines the coefficients which arise in binomial expansions.

Notice that the entire right diagonal of Pascal's triangle corresponds to the coefficient of y < sup > n </ sup > in these binomial expansions, while the next diagonal corresponds to the coefficient of xy < sup > n − 1 </ sup > and so on.

To see how the binomial theorem relates to the simple construction of Pascal's triangle, consider the problem of calculating the coefficients of the expansion of ( x + 1 )< sup > n + 1 </ sup > in terms of the corresponding coefficients of ( x + 1 )< sup > n </ sup > ( setting y

When divided by 2 < sup > n </ sup >, the nth row of Pascal's triangle becomes the binomial distribution in the symmetric case where p = 1 / 2.

Specific coefficients arise in mathematical identities, such as the binomial theorem which involves binomial coefficients ; these particular coefficients are tabulated in Pascal's triangle.

An inductive proof for arithmetic sequences was introduced in the Al-Fakhri ( 1000 ) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle.

In the Treatise he wrote on the triangular array of binomial coefficients known as Pascal's triangle.

* Blaise Pascal publishes his Traité du triangle arithmétique in which he describes a convenient tabular presentation for binomial coefficients, now called Pascal's triangle.

: can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle.

The earliest explicit depictions of a triangle of binomial coefficients occur in the 10th century in commentaries on the Chandas Shastra, an Ancient Indian book on Sanskrit prosody written by Pingala in or before the 2nd century BC.

Several theorems related to the triangle were known, including the binomial theorem.

Each performance of an n-trial binomial experiment results in some whole number from 0 through N as the value of the random variable X, where Af.

where each is a specific positive integer known as binomial coefficient.

The same number however occurs in many other mathematical contexts, where it is denoted by ( often read as " n choose k "); notably it occurs as coefficient in the binomial formula, hence its name binomial coefficient.

where is the binomial coefficient.

Similarly if what were previously thought to be two distinct species are demoted to a lower rank, such as subspecies, where possible the second part of the binomial name is retained as the third part of the new name.

For the special case where r is an integer, the negative binomial distribution is known as the Pascal distribution.

where the latter, which is immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the multiplicative formula of the binomial coefficient.

where we have used the following < span id =" theorem "> theorem </ span > involving binomial coefficients:

where is a binomial coefficient.

In the special case where there are only two cells in the table, the expected values follow a binomial distribution,

where p ≤ x runs over primes, and the binomial coefficients.

The American White Ibis was one of the many bird species originally described by Carl Linnaeus in the 1758 10th edition of his Systema Naturae, where it was given the binomial name of Scolopax albus.

The White Stork was one of the many bird species originally described by Linnaeus in the landmark 1758 10th edition of his Systema Naturae, where it was given the binomial name of Ardea ciconia.

The Black Stork was one of the many species originally described by Linnaeus in the landmark 1758 10th edition of his Systema Naturae, where it was given the binomial name of Ardea nigra.

* tautonym: a binomial or scientific name in the taxonomy of living things in which the generic and specific names are the same, such as Gorilla gorilla ; a scientific name in which the specific name is repeated, such as Homo sapiens sapiens as distinct from Homo sapiens neanderthalensis ; a noun component that is repeated, such as aye-aye or tom-tom ; a personal name where both forename and surname are identical, such as Francis Francis

where is a binomial coefficient.

Thus he published in 1790, a Index Ornithologicus where he specified a binomial name for all the species which he had previously described.