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The 58 pages that link to Binomial coefficient, each with the reason it gives.

• CombinatoricsRelated: These coefficients encode many fundamental selection counts and identities.

• FactorialRelated: Its factorial formula is n! divided by k!(n − k)!.

• Binomial distributionRelated: It counts which trials succeed when the total success count is fixed.

• Power setRelated: It counts the power-set members having each particular size.

• Binomial theoremBroader topic: Each expansion term is weighted by the number of ways to choose which factors contribute the second summand.

• Finite differenceRelated: Binomial coefficients appear in the formula for differences of higher order.

• Fibonacci sequenceRelated: Sums along diagonals of Pascal's triangle produce Fibonacci numbers.

• Inclusion–exclusion principleRelated: It counts how many subsets of conditions contain a given element.

• CoefficientBroader topic: These coefficients weight terms in powers of a binomial, as in the binomial theorem.

• Pascal's triangleBroader topic: Each entry in row n is a binomial coefficient, generated from its two neighbors in the preceding row.

• MultisetRelated: Counting multisets of fixed size leads to combinations with repetition.

• Catalan numberNarrower topic: The closed form for Catalan numbers is expressed using binomial coefficients.

• Birthday ProblemRelated: It counts the pairs of people whose birthdays could match.

• Bertrand's postulateRelated: Central binomial coefficients encode prime factors that force a prime between n and 2n.

• Stirling numbers of the second kindRelated: It appears in the identity S(n,k)=Σⱼ₌₀ᵏ⁻¹ binom(n−1,j)S(n−1−j,k−1).

• Exterior algebraRelated: For an n-dimensional space, the degree-k exterior power has dimension n choose k.

• Jacob BernoulliRelated: Binomial coefficients count outcome sequences in repeated two-outcome experiments.

• CombinationRelated: It counts combinations of a given size from a set of distinct objects.

• Combinatorial proofBroader topic: Many identities become transparent when each side counts selections with different distinguished features.

• DiagonalRelated: Choosing two vertices gives the total candidate segments before boundary edges are excluded.

• Hypergeometric distributionBroader topic: Combinations count favorable samples and all possible samples in its probability formula.

• Negative binomial distributionRelated: It counts the ways failures can appear before the final required success.

• Stars and barsRelated: It gives the formula for choosing separator positions among stars and bars.

• Falling factorialRelated: Dividing a falling factorial by n! gives the binomial coefficient when n is a nonnegative integer.

• Multinomial coefficientRelated: Repeated binomial choices decompose a multinomial count into successive selections.

• Gaussian binomial coefficientCompared with: It counts subsets rather than linear subspaces and is recovered by setting the parameter to one.

• Vandermonde's identityBroader topic: Each term in the identity is a product of two such counts.

• Yang HuiBroader topic: Its coefficients form the triangular arrangement associated with Yang Hui.

• Bell numberCompared with: Both count finite combinatorial choices, but Bell numbers allow arbitrary groupings rather than one chosen subset.

• Erdős–Ko–Rado theoremRelated: It gives the exact maximum number of sets in the theorem.

• Triangular numberRelated: The nth triangular number equals the number of ways to choose two objects from n + 1.

• Binomial (algebra)Compared with: Despite the shared name, it is a counting quantity, not a two-term polynomial.

• Legendre's formulaRelated: Subtracting three factorial valuations gives each prime's exponent in a binomial coefficient.

• Newton's identitiesRelated: Equivalent identity conventions use binomial coefficients to normalize sums.

• Pascal's ruleNarrower topic: Pascal's rule is a recurrence relating binomial coefficients with neighboring indices.

• Turán's theoremRelated: It counts possible edges and appears in exact forms of the extremal bound.

• Hockey-stick identityNarrower topic: Every term in the identity is a binomial coefficient, and the result is another one.

• Jia XianBroader topic: These numbers form the triangular arrangement associated with Jia Xian.

• Sperner's theoremRelated: The theorem's bound is the largest binomial coefficient among the subset-lattice ranks.

• Cauchy–Binet formulaRelated: The sum ranges over subsets of intermediate indices, whose number is a binomial coefficient.

• Erdős–Rado theoremRelated: Partition notation uses brackets to indicate which finite subsets are colored.

• Gilbert–Varshamov boundRelated: For binary strings, each Hamming sphere’s size is a binomial coefficient.

• Abel's binomial theoremRelated: These coefficients weight the terms in Abel’s sum.

• Bertrand's ballot theoremRelated: It counts all possible placements of the p votes among the p+q positions.

• Bohr–Mollerup theoremRelated: The gamma function supplies a continuous extension of factorial-based coefficient formulas.

• Bonse's inequalityRelated: Binomial-coefficient estimates provide a standard route to bounds involving products of primes.

• Freshman's dreamBroader topic: The intermediate coefficients are multiples of p and therefore zero in characteristic p.

• Hamming boundRelated: The volume of a Hamming ball is a sum of binomial coefficients for a binary alphabet.

• NonogramRelated: It helps count how many arrangements can fit a line’s clue and remaining gaps.

• Agrawal's conjectureBroader topic: Intermediate coefficients determine whether the polynomial identity holds.