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  2. Binomial theorem - Wikipedia

    en.wikipedia.org/wiki/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 ...

  3. Power rule - Wikipedia

    en.wikipedia.org/wiki/Power_rule

    The power rule for differentiation was derived by Isaac Newton and Gottfried Wilhelm Leibniz, each independently, for rational power functions in the mid 17th century, who both then used it to derive the power rule for integrals as the inverse operation. This mirrors the conventional way the related theorems are presented in modern basic ...

  4. Power law - Wikipedia

    en.wikipedia.org/wiki/Power_law

    The distributions of a wide variety of physical, biological, and human-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, [2] cloud sizes, [3] the foraging pattern of various species, [4] the sizes of activity patterns of neuronal populations, [5] the frequencies of words in most languages ...

  5. Trinomial expansion - Wikipedia

    en.wikipedia.org/wiki/Trinomial_expansion

    In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by The expansion is given by ( a + b + c ) n = ∑ i , j , k i + j + k = n ( n i , j , k ) a i b j c k , {\displaystyle (a+b+c)^{n}=\sum _{{i,j,k} \atop {i+j+k=n}}{n \choose i,j,k}\,a^{i}\,b^{\;\!j}\;\!c^{k},}

  6. Sums of powers - Wikipedia

    en.wikipedia.org/wiki/Sums_of_powers

    In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.

  7. Multinomial theorem - Wikipedia

    en.wikipedia.org/wiki/Multinomial_theorem

    In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials .

  8. Binomial coefficient - Wikipedia

    en.wikipedia.org/wiki/Binomial_coefficient

    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 . Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written ( n k ) . {\displaystyle {\tbinom {n}{k}}.}

  9. Legendre's formula - Wikipedia

    en.wikipedia.org/wiki/Legendre's_formula

    Legendre's formula can be used to prove Kummer's theorem. As one special case, it can be used to prove that if n is a positive integer then 4 divides ( 2 n n ) {\displaystyle {\binom {2n}{n}}} if and only if n is not a power of 2.