Binomial theorem and pascals triangle

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Binomial theorem and pascals triangle

Triangular array of the binomial coefficients in mathematics. Formula [ edit ]. History [ edit ]. Binomial expansions [ edit ]. Combinations [ edit ]. Relation to binomial distribution and convolutions [ edit ]. Patterns and properties [ edit ]. Rows [ edit ]. Diagonals [ edit ]. Calculating a row or diagonal by itself [ edit ]. Overall patterns and properties [ edit ].

Construction as matrix exponential [ edit ]. See also: Pascal matrix. Construction of Clifford algebra using simplices [ edit ]. Relation to geometry of polytopes [ edit ]. This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed.

October Learn how and when to remove this message. Number of elements of simplices [ edit ]. Number of elements of hypercubes [ edit ]. Counting vertices in a cube by distance [ edit ]. Extensions [ edit ]. To higher dimensions [ edit ]. To complex numbers [ edit ]. To arbitrary bases [ edit ]. See also [ edit ]. Bean machineFrancis Galton's "quincunx" Bell triangle Bernoulli's triangle Binomial expansion Cellular automata Euler triangle Floyd's triangle Gaussian binomial coefficient Hockey-stick identity Leibniz harmonic triangle Multiplicities of entries in Pascal's triangle Singmaster's conjecture Pascal matrix Pascal's pyramid Pascal's simplex Proton NMRone application of Pascal's triangle Star of David theorem Trinomial expansion Trinomial triangle Polynomials calculating sums of powers of arithmetic progressions.

References [ edit ]. Cambridge University Library: the great collections. Cambridge University Press. ISBN Bibcode : ehst. Other, lost works of al-Karaji's are known to have dealt with inderterminate algebra, arithmetic, inheritance algebra, and the construction of buildings. Another contained the first known explanation of the arithmetical Pascal's triangle; the passage in question survived through al-Sama'wal's Bahir twelfth century which heavily drew from the Badi.

However, the use of binomial coefficients by Islamic mathematicians of the eleventh century, in a context which had deep roots in Islamic mathematics, suggests strongly the table was a local discovery - most probably of al-Karaji. Indian Journal of History of Science. Translated by S. Ein Beitrag zur indischen Mathematik". Zeitschrift fur Indologie und Iranistik.

External links [ edit ]. Blaise Pascal. Pascaline Pascal's law Pascal's theorem Pascal's triangle Pascal's wager. For example, there will only be one term x ncorresponding to choosing x from each binomial. For a given kthe following are proved equal in succession:. Induction yields another proof of the binomial theorem. AroundIsaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers.

The same generalization also applies to complex exponents. In this generalization, the finite sum is replaced by an infinite series. In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials. This agrees with the usual definitions when r is a nonnegative integer.

For other values of rthe series typically has infinitely many nonzero terms. The generalized binomial theorem can be extended to the case where x and y are complex numbers. The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is. For each term in the expansion, the exponents must add up to n.

When working in more dimensions, it is often useful to deal with products of binomial expressions. Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting k objects out of n without replacement combinationswere of interest to ancient Indian mathematicians. The Jain Bhagavati Sutra c. In Europe, descriptions of the construction of Pascal's triangle can be found as early as Jordanus de Nemore 's De arithmetica 13th century.

For the complex numbers the binomial theorem can be combined with de Moivre's formula to yield multiple-angle formulas for the sine and cosine. Using the binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for cos nx and sin nx. Applying the binomial theorem to this expression yields the usual infinite series for e.

Indeed, since each term of the binomial expansion is an increasing function of nit follows from the monotone convergence theorem for binomial theorem and pascals triangle that the sum of this infinite series is equal to e. The binomial theorem is closely related to the probability mass function of the negative binomial distribution.

Contents move to sidebar hide. Article Talk. Read Edit View history. Tools Tools. Download as PDF Printable version. In other projects. Wikimedia Commons Wikidata item. Algebraic expansion of powers of a binomial. That is. General Rule :. In pascal expansion, we must have only 'a' in the first term, only 'b' in the last term and 'ab' in all other middle terms.

This rule is not only applicable for power '4'. It has been clearly explained below. Now we have to follow the steps given below. Then, the first term will be a 4. In the second term, we have to take both 'a' and 'b'. For 'a', we have to take exponent '1' less than the exponent of 'a' in the previous term. For 'b', we have to take exponent '1'.

Then, the second term will be a 3 b. In the third term also, we have to take both 'a' and 'b'. For 'b', we have to take exponent '2'. Then, the second term will be a 2 b 2.