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The 36 pages that link to Binomial theorem, each with the reason it gives.

• Binomial coefficientRelated: The coefficient's algebraic role follows directly from this expansion.

• Mathematical inductionRelated: Induction on the exponent provides a direct proof of its expansion formula.

• Generating functionRelated: Binomial expansions give standard generating functions and coefficient identities.

• FactorialRelated: Factorials in the coefficients make the expansion's combinatorial weights precise.

• Binomial distributionRelated: Its expansion supplies the normalization that makes the probabilities sum to one.

• Power setRelated: Summing the counts of subsets of each size gives (1 + 1)ⁿ = 2ⁿ.

• Taylor's theoremRelated: Its coefficient pattern matches Taylor expansions of powers and products.

• Inclusion–exclusion principleRelated: Binomial coefficients encode how often each element appears in the alternating corrections.

• Omar KhayyamRelated: Khayyam used a numerical triangle to derive binomial coefficients, anticipating later formulations.

• Pascal's triangleRelated: Each row supplies the coefficients for expanding a power of a two-term sum.

• Frobenius endomorphismRelated: Modulo p, the intermediate binomial coefficients vanish when the exponent is p.

• Cauchy productRelated: Repeated Cauchy products of the series for 1+x produce binomial coefficients.

• CombinationRelated: Its coefficients count how many ways terms contribute to each power.

• Combinatorial proofRelated: Choosing terms from repeated factors gives a combinatorial explanation for its coefficients.

• Perfect squareRelated: Expanding (n + 1)² explains why successive squares differ by odd numbers.

• Falling factorialRelated: The falling-factorial binomial identity extends the familiar additive pattern to polynomial sequences.

• Bernoulli's inequalityRelated: For integer exponents, the expansion exposes the nonnegative terms behind the bound when x≥0.

• Vandermonde's identityRelated: Multiplying two binomial-theorem expansions produces Vandermonde's coefficient sum.

• Binomial (algebra)Related: It generalizes the expansion of a binomial raised to any nonnegative integer power.

• Pascal's ruleRelated: Pascal's recurrence generates the coefficients used in successive binomial expansions.

• Al-KarajiRelated: His arithmetic of powers included rules related to expansions of binomial expressions.

• Hockey-stick identityRelated: It connects the coefficients in the identity to polynomial expansions.

• Jia XianRelated: The triangular coefficients associated with Jia Xian encode these expansions.

• Freshman's dreamRelated: Its expansion shows which intermediate terms must vanish for the identity to hold.

• History of combinatoricsRelated: Its coefficients connect algebraic expansion with counting subsets and arrangements.

• Lifting-the-exponent lemmaRelated: Binomial expansions help explain why increasing an exponent can raise a prime-adic valuation.

• Agrawal's conjectureRelated: Expanding (x+a)^n exposes the coefficients that must vanish modulo n.

• Combinatorial principlesRelated: Its coefficients count how many ways each term arises in a product.

• Twelvefold wayRelated: Its coefficients appear when each distinguishable object independently chooses a box.

• Wolstenholme's theoremRelated: It connects the coefficient in the theorem to familiar identities and algebraic expansions.