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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. Binomial coefficient - Wikipedia

    en.wikipedia.org/wiki/Binomial_coefficient

    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.

  4. Binomial series - Wikipedia

    en.wikipedia.org/wiki/Binomial_series

    The binomial series is therefore sometimes referred to as Newton's binomial theorem. Newton gives no proof and is not explicit about the nature of the series. Later, on 1826 Niels Henrik Abel discussed the subject in a paper published on Crelle's Journal, treating notably questions of convergence. [4]

  5. Proofs of Fermat's little theorem - Wikipedia

    en.wikipedia.org/wiki/Proofs_of_Fermat's_little...

    Proof of Lemma. We consider the binomial coefficient when the exponent is a prime p: =!! )! The binomial coefficients are all integers. ... This multinomial expansion ...

  6. Multinomial theorem - Wikipedia

    en.wikipedia.org/wiki/Multinomial_theorem

    Multinomial coefficient as a product of binomial coefficients, counting the permutations of the letters of MISSISSIPPI. The multinomial coefficient (, …,) is also the number of distinct ways to permute a multiset of n elements, where k i is the multiplicity of each of the i th element. For example, the number of distinct permutations of the ...

  7. Pascal's rule - Wikipedia

    en.wikipedia.org/wiki/Pascal's_rule

    In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients.It states that for positive natural numbers n and k, + = (), where () is a binomial coefficient; one interpretation of the coefficient of the x k term in the expansion of (1 + x) n.

  8. Binomial approximation - Wikipedia

    en.wikipedia.org/wiki/Binomial_approximation

    The binomial approximation for the square root, + + /, can be applied for the following expression, + where and are real but .. The mathematical form for the binomial approximation can be recovered by factoring out the large term and recalling that a square root is the same as a power of one half.

  9. Lucas's theorem - Wikipedia

    en.wikipedia.org/wiki/Lucas's_theorem

    Lucas's theorem can be generalized to give an expression for the remainder when () is divided by a prime power p k.However, the formulas become more complicated. If the modulo is the square of a prime p, the following congruence relation holds for all 0 ≤ s ≤ r ≤ p − 1, a ≥ 0, and b ≥ 0.