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Statements

Subject Item: dbr:Campanology
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Subject Item: dbr:Bell_polynomials
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Subject Item: dbr:Primitive_recursive_function
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Subject Item: dbr:Primorial
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Subject Item: dbr:Proofs_That_Really_Count
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Subject Item: dbr:Proofs_of_Fermat's_little_theorem
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Subject Item: dbr:Python_(programming_language)
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Subject Item: dbr:Qualitative_variation
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Subject Item: dbr:Rosetta_Code
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Subject Item: dbr:Rubik's_Cube
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Subject Item: dbr:Rubik's_family_cubes_of_varying_sizes
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Subject Item: dbr:Entropy_of_mixing
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Subject Item: dbr:Enumerative_combinatorics
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Subject Item: dbr:Memoization
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Subject Item: dbr:Method_ringing
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Subject Item: dbr:Multifactorial
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Subject Item: dbr:M-94
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Subject Item: dbr:Prime_gap
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Subject Item: dbr:Thomas_Jarrett
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Subject Item: dbr:V-Cube_6
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Subject Item: dbr:Benford's_law
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Subject Item: dbr:Bessel_function
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Subject Item: dbr:Beta_function
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Subject Item: dbr:Binomial_coefficient
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Subject Item: dbr:Binomial_theorem
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Subject Item: dbr:Birthday_problem
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Subject Item: dbr:Boltzmann's_entropy_formula
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Subject Item: dbr:Borel_summation
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Subject Item: dbr:Dc_(computer_program)
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Subject Item: dbr:Denotational_semantics
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Subject Item: dbr:Determinant
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Subject Item: dbr:Alignments_of_random_points
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Subject Item: dbr:Arbitrary-precision_arithmetic
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Subject Item: dbr:Hosoya_index
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Subject Item: dbr:Hyperfactorial
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Subject Item: dbr:Jordan–Pólya_number
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Subject Item: dbr:Bertrand's_postulate
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Subject Item: dbr:Bhargava_factorial
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Subject Item: dbr:List_of_integer_sequences
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Subject Item: dbr:List_of_mathematical_symbols_by_subject
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Subject Item: dbr:List_of_numbers
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Subject Item: dbr:List_of_prime_numbers
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Subject Item: dbr:List_of_sums_of_reciprocals
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Subject Item: dbr:Pentatope_number
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Subject Item: dbr:Permutation
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Subject Item: dbr:Riemann–Liouville_integral
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Subject Item: dbr:Unary_operation
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Subject Item: dbr:Volume_of_an_n-ball
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Subject Item: dbr:Derangement
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Subject Item: dbr:Desmos
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Subject Item: dbr:Double_exponential_function
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Subject Item: dbr:Index_of_a_subgroup
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Subject Item: dbr:Index_of_genetics_articles
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Subject Item: dbr:Integer_sequence
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Subject Item: dbr:Interpolation_sort
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Subject Item: dbr:John_21
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Subject Item: dbr:Real_analysis
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Subject Item: dbr:List_of_inequalities
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Subject Item: dbr:List_of_logarithmic_identities
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Subject Item: dbr:List_of_mathematical_functions
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Subject Item: dbr:List_of_permutation_topics
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Subject Item: dbr:List_of_representations_of_e
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Subject Item: dbr:The_Penguin_Dictionary_of_Curious_and_Interesting_Numbers
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Subject Item: dbr:Pentellated_8-simplexes
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Subject Item: dbr:Point_group
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Subject Item: dbr:Stanley_symmetric_function
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Subject Item: dbr:Stat-Ease
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Subject Item: dbr:Uniform_6-polytope
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Subject Item: dbr:!_(disambiguation)
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Subject Item: dbr:%22Hello,_World!%22_program
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Subject Item: dbr:0
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Subject Item: dbr:1,000,000,000
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Subject Item: dbr:10
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Subject Item: dbr:10,000,000
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Subject Item: dbr:100,000
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Subject Item: dbr:119_(number)
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Subject Item: dbr:11_(number)
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Subject Item: dbr:120_(number)
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Subject Item: dbr:145_(number)
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Subject Item: dbr:153_(number)
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Subject Item: dbr:15_(number)
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Subject Item: dbr:154_(number)
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Subject Item: dbr:158_(number)
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Subject Item: dbr:170_(number)
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Subject Item: dbr:Combinatorial_explosion
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Subject Item: dbr:Complex_polytope
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Subject Item: dbr:Anagram
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Subject Item: dbr:Mega_Millions
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Subject Item: dbr:Rust_(programming_language)
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Subject Item: dbr:Gaussian_integral
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Subject Item: dbr:Gaussian_q-distribution
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Subject Item: dbr:Generating_function_transformation
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Subject Item: dbr:Genetic_drift
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Subject Item: dbr:Optimality_Theory
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Subject Item: dbr:Paramorphism
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Subject Item: dbr:Table_of_mathematical_symbols_by_introduction_date
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Subject Item: dbr:Ordered_Bell_number
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Subject Item: dbr:Sefer_Yetzirah
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Subject Item: dbr:Pyraminx_Duo
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Subject Item: dbr:Q-Pochhammer_symbol
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Subject Item: dbr:Q-analog
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Subject Item: dbr:Trailing_zero
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Subject Item: dbr:Scrabble_variants
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Subject Item: dbr:1808_in_science
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Subject Item: dbr:Christian_Kramp
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dbo:knownFor: dbr:Factorial

Subject Item: dbr:Clean_(programming_language)
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Subject Item: dbr:Ellipsis
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Subject Item: dbr:Emmy_Noether
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Subject Item: dbr:Function_(mathematics)
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Subject Item: dbr:Gamma
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Subject Item: dbr:Gentzen's_consistency_proof
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Subject Item: dbr:Glaisher–Kinkelin_constant
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Subject Item: dbr:Glossary_of_mathematical_symbols
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Subject Item: dbr:Grand_canonical_ensemble
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Subject Item: dbr:Modern_Arabic_mathematical_notation
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Subject Item: dbr:Multi-index_notation
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Subject Item: dbr:Multinomial_theorem
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Subject Item: dbr:Multiplication
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Subject Item: dbr:Mxparser
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Subject Item: dbr:Constant-recursive_sequence
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Subject Item: dbr:Convergent_series
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Subject Item: dbr:Corecursion
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Subject Item: dbr:Cribbage_(pool)
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Subject Item: dbr:Cross-ratio
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Subject Item: dbr:Equidissection
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Subject Item: dbr:Uniform_9-polytope
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Subject Item: dbr:Precondition
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Subject Item: dbr:TI-BASIC
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Subject Item: dbr:Arity
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Subject Item: dbr:Lenstra_elliptic-curve_factorization
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Subject Item: dbr:Levi-Civita_symbol
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Subject Item: dbr:Lisp_(programming_language)
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Subject Item: dbr:Logarithm
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Subject Item: dbr:Lottery_mathematics
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Subject Item: dbr:Louis_François_Antoine_Arbogast
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Subject Item: dbr:Lua_(programming_language)
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Subject Item: dbr:Ludwig_Boltzmann
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Subject Item: dbr:ML_(programming_language)
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Subject Item: dbr:Cain's_Jawbone
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Subject Item: dbr:Cake_number
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Subject Item: dbr:Calculator_spelling
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Subject Item: dbr:Smalltalk
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Subject Item: dbr:Stirling's_approximation
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Subject Item: dbr:Combination
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Subject Item: dbr:Comparison_sort
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Subject Item: dbr:Complete_spatial_randomness
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Subject Item: dbr:Completely_randomized_design
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Subject Item: dbr:Computational_complexity_of_mathematical_operations
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Subject Item: dbr:Empty_product
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Subject Item: dbr:Freshman's_dream
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Subject Item: dbr:Half-integer
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Subject Item: dbr:Hamiltonian_decomposition
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Subject Item: dbr:Harmonic_series_(mathematics)
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Subject Item: dbr:Joseph_Ser
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Subject Item: dbr:Parity_of_a_permutation
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Subject Item: dbr:Prefix_sum
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Subject Item: dbr:Uwe_Storch
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Subject Item: dbr:Postcondition
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Subject Item: dbr:1
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Subject Item: dbr:Ball_(mathematics)
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Subject Item: dbr:5040_(number)
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gold:hypernym: dbr:Factorial

Subject Item: dbr:69_(number)
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Subject Item: dbr:700_(number)
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Subject Item: dbr:720_(number)
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Subject Item: dbr:79_(number)
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Subject Item: dbr:880_(number)
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Subject Item: dbr:Brocard's_problem
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Subject Item: dbr:Bullvalene
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Subject Item: dbr:Transcendental_number
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Subject Item: dbr:Travelling_salesman_problem
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Subject Item: dbr:Triangular_number
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Subject Item: dbr:Twelve-tone_technique
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Subject Item: dbr:WebAssembly
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Subject Item: dbr:Windows_Calculator
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Subject Item: dbr:Do_while_loop
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Subject Item: dbr:GROW_(series)
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Subject Item: dbr:Gödel_numbering_for_sequences
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Subject Item: dbr:Hadamard's_gamma_function
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Subject Item: dbr:Harshad_number
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Subject Item: dbr:Hash_function
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Subject Item: dbr:Haskell_features
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Subject Item: dbr:Helicopter_Cube
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Subject Item: dbr:Irrationality_sequence
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Subject Item: dbr:Lattice_of_stable_matchings
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Subject Item: dbr:Logarithmically_convex_function
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Subject Item: dbr:3000_(number)
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Subject Item: dbr:400_(number)
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Subject Item: dbr:Airy_zeta_function
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Subject Item: dbr:D_(programming_language)
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Subject Item: dbr:Erlang_(programming_language)
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Subject Item: dbr:Euclid's_theorem
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Subject Item: dbr:Euler–Maclaurin_formula
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Subject Item: dbr:Exclamation_mark
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Subject Item: dbr:Factor_(programming_language)
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Subject Item: dbr:Factorials
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Subject Item: dbr:Hylomorphism_(computer_science)
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Subject Item: dbr:Characterizations_of_the_exponential_function
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Subject Item: dbr:Recursion_(computer_science)
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Subject Item: dbr:Factorial
rdf:type: owl:Thing
rdfs:label: Fattoriale Факторіал Faktorial Faktorialo Fakultät (Mathematik) عاملي Silnia Factorial Faktorial Факториал Παραγοντικό 階乘 Factorielle Factorial 계승 階乗 Fakultet (matematik) Faculteit (wiskunde) Fatorial Factorial Faktoriál
rdfs:comment: Στα μαθηματικά τo παραγοντικό ενός φυσικού αριθμού ν συμβολίζεται με ν!, διαβάζεται νι παραγοντικό, και είναι το γινόμενο όλων των θετικών ακεραίων μικρότερων ή ίσων με ν: ν! = 1 ∙ 2 ∙ 3 ∙ ... ∙ ν Για παράδειγμα, 2!=1·2= 2 3!=1·2·3= 6 4!=1·2·3·4= 24 5!=1·2·3·4·5= 120 8!=1·2·3·4·5·6·7·8= 40.320 10!=1·2·3·4·5·6·7·8·9·10= 3.628.800 12!=1·2·3·4·5·6·7·8·9·10·11·12= 479.001.600 Το παραγοντικό ενός αριθμού ν εκφράζει και το πλήθος των δυνατών μεταθέσεων των ν στοιχείων ενός συνόλου, δηλαδή το πλήθος των διαφορετικών τρόπων με τους οποίους μπορούμε να βάλουμε σε μια σειρά τα ν στοιχεία ενός συνόλου. 5!=120 数学において非負整数 n の階乗（かいじょう、英: factorial）n ! は、1 から n までの全ての整数の積である。例えば、である。空積の規約のもと 0! = 1 と定義する。階乗は数学の様々な場面に出現するが、特に組合せ論、代数学、解析学などが著しい。階乗の最も基本的な出自は n 個の相異なる対象を1列に並べる方法（対象の置換）の総数が n! 通りであるという事実である。階乗の定義は、最も重要な性質を残したまま、非整数を引数とする函数にすることができる。そうすれば解析学における著しい手法などの進んだ数学を利用できるようになる。 Dalam matematika, Faktorial dari bilangan bulat positif dari n yang dilambangkan dengan n!, adalah produk dari semua bilangan bulat positif yang kurang dari atau sama dengan n: Sebagai contoh, Nilai 0! adalah 1, menurut konvensi untuk . Operasi faktorial digunakan sebagai bidang matematika, terutama di kombinatorik, aljabar, dan analisis matematika. Penggunaannya yang paling dasar menghitung kemungkinan urutan dan permutasi dari n yang berada di objekk yang berbeda. El factorial de un entero positivo n, el factorial de n o n factorial se define en principio como el producto de todos los números enteros positivos desde 1 (es decir, los números naturales) hasta n. Por ejemplo: La operación de factorial aparece en muchas áreas de las matemáticas, particularmente en combinatoria y análisis matemático.De manera fundamental el factorial de n representa el número de formas distintas de ordenar n objetos distintos (elementos sin repetición). Este hecho ha sido conocido desde hace varios siglos, en el siglo XII por los estudiosos hindúes. ( 계승(繼承)에 대해서는 왕위 계승 문서를 참고하십시오.) 수학에서, 자연수의 계승 또는 팩토리얼(階乘, 문화어: 차례곱, 영어: factorial)은 그 수보다 작거나 같은 모든 양의 정수의 곱이다. n이 하나의 자연수일 때, 1에서 n까지의 모든 자연수의 곱을 n에 상대하여 이르는 말이다. 기호는 을 쓰며 팩토리얼이라고 읽는다. 공식적이지는 않지만 한국 사람들 사이에서 팩토리얼을 줄여서 팩이라고 읽기도 한다. في الرياضيات، المضروب أو العاملي لعدد صحيح طبيعي n، والذي يكتب ، والذي يقرأ "عاملي n"، هو جداء كل الأعداد الطبيعية (الأعداد الصحيحة الموجبة قطعاً) المساوية أو الأصغر من n، ما عدا الصفر. فيما يلي مثال 5 عاملي: و تعريف العاملي على شكل جداء يترتب عنه كون ذلك أن 0! جداء مفرغ، وبمعنى آخر مختصر أي عدد مضروب في صفر يساوي صفر في عملية الضرب. يمكن لتعريف دالة عاملي أن يمدد إلى أعداد غير صحيحة بدون المساس بخصائص هذه الدالة. هذه العملية تستلزم تقنيات متطورة في الرياضيات وخصوصا تلك المستقاة من التحليل الرياضي. Факториа́л — функция, определённая на множестве неотрицательных целых чисел. Название происходит от лат. factorialis — действующий, производящий, умножающий; обозначается , произносится эн факториа́л. Факториал натурального числа определяется как произведение всех натуральных чисел от 1 до включительно: . Например, . Для принимается в качестве соглашения, что . Факториал активно используется в различных разделах математики: комбинаторике, математическом анализе, теории чисел, функциональном анализе и др. In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: For example,The value of 0! is 1, according to the convention for an empty product. Die Fakultät (manchmal, besonders in Österreich, auch Faktorielle genannt) ist in der Mathematik eine Funktion, die einer natürlichen Zahl das Produkt aller natürlichen Zahlen (ohne Null) kleiner und gleich dieser Zahl zuordnet. Sie wird durch ein dem Argument nachgestelltes Ausrufezeichen („!“) abgekürzt. Diese Notation wurde erstmals 1808 von dem elsässischen Mathematiker Christian Kramp (1760–1826) verwendet, der um 1798 auch die Bezeichnung faculté „Fähigkeit“ dafür einführte. Na matemática, o fatorial (AO 1945: factorial) de um número natural n, representado por n!, é o produto de todos os inteiros positivos menores ou iguais a n. A notação n! foi introduzida por Christian Kramp em 1808. Edozein n zenbakiaren faktoriala, n zenbaki arrunta izanik, 1 eta n artean dauden zenbaki natural guztien biderkaduraren emaitza da. Adibidez: n! notazioa matematikariak sortu zuen. V matematice je faktoriál čísla n (značeno pomocí vykřičníku: n!) číslo, rovné součinu všech kladných celých čísel menších nebo rovných n, pokud je n kladné, a rovno 1 pro n = 0. Značení n! vyslovujeme jako „n faktoriál“. Toto značení zavedl Christian Kramp v roce 1808. Факторіал натурального числа — добуток натуральних чисел від одиниці до включно, позначається !. . За означенням , згідно з конвенцією для .. При великих наближене значення факторіала можна обчислити за формулою Стірлінга. Факторіал дорівнює кількості перестановок з елементів. En matemàtiques, el factorial d'un enter no negatiu , denotat per (en alguns llibres antics es pot trobar denotat per ), és el producte de tots els nombres enters positius inferiors o iguals a . Per exemple, El valor de és 1, d'acord amb la convenció d'un producte buit. La notació va ser introduïda pel matemàtic francès Christian Kramp el 1808. La definició de la funció factorial també es pot ampliar a arguments no enters, tot conservant les seves propietats més importants; això implica matemàtiques més avançades, especialment tècniques d'anàlisi matemàtica. En la matematiko, faktorialo de natura nombro n estas la produto de la pozitivaj entjeroj malpli aŭ egalaj al n. Oni signas ĝin per n!, kion oni prononcas no faktoriale laŭ Christian Kramp. En mathématiques, la factorielle d'un entier naturel n est le produit des nombres entiers strictement positifs inférieurs ou égaux à n. Cette opération est notée avec un point d'exclamation, n!, ce qui se lit soit « factorielle de n », soit « factorielle n », soit « n factorielle ». Cette notation a été introduite en 1808 par Christian Kramp. La factorielle joue un rôle important en algèbre combinatoire parce qu'il y a n! façons différentes de permuter n objets. Elle apparaît dans de nombreuses formules en mathématiques, comme la formule du binôme et la formule de Taylor. In matematica, si definisce fattoriale di un numero naturale , indicato con , il prodotto dei numeri interi positivi minori o uguali a tale numero. In formula: per la convenzione del prodotto vuoto si definisce inoltre . La generalizzazione analitica del fattoriale è nota con il nome di funzione gamma di Eulero. La notazione con il punto esclamativo è stata introdotta nel 1807 da Christian Kramp, mentre il nome fattoriale era stato coniato pochi anni prima, nel 1800 da Antoine Arbogast. La sequenza dei fattoriali compare nella On-Line Encyclopedia of Integer Sequences (OEIS) come sequenza . Fakultet är en funktion inom matematiken. För ett heltal större än noll är fakulteten lika med produkten av alla heltal från 1 upp till och med talet självt. De faculteit van een natuurlijk getal , genoteerd als (n faculteit), is het product van de getallen tot en met : Recursief geldt dus voor de faculteit: Voor bijvoorbeeld is: In overeenstemming met de definitie van het lege product is afgesproken dat De faculteitsfunctie groeit snel, zelfs sneller dan een exponentiële functie. De eerste 20 waarden, met nul, staan hiernaast. Het aantal decimalen van n! , met n > 1 , is gelijk aan 10log 1 + ... + 10log n naar boven afgerond. Voor n = 1000 komt het aantal decimalen op 2568. 在數學中，正整数的階乘（英語：Factorial）是所有小於等於該數的正整數的積，計為n!，例如5的階乘表示為5!，其值為120：並定義，1的階乘1!和0的階乘0!都為1，其中0的階乘表示一個空積。 1808年，基斯頓·卡曼引進這個表示法：，符號表示連續乘積，亦即n!=1×2×3×...×n。階乘亦可以遞迴方式定義：0!=1，n!=(n-1)!×n。除了自然數之外，階乘亦可定義于整個實數（負整數除外），其与伽瑪函數的关系为：階乘應用在許多數學領域中，最常應用在組合學、代數學和数学分析中。在組合學中，階乘代表的意義為n個相異物件任意排列的數量，例如前述例子，其代表了5個相異物件共有120種排列法。在正整數的情形下，n的階乘又可以稱為n的排列數。 Silnia liczby naturalnej n – iloczyn wszystkich liczb naturalnych dodatnich nie większych niż . Zapis itd. odczytujemy „n silnia”, „dwa silnia” itd.
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dcterms:subject: dbc:Factorial_and_binomial_topics dbc:Combinatorics dbc:Unary_operations dbc:Gamma_and_related_functions
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dbo:thumbnail: n16:Mplwp_factorial_stirling_loglog2.svg?width=300
dbp:cs1Dates: ly
dbp:date: December 2021
dbp:id: p/f038080
dbp:title: Factorial
dbp:urlname: Factorial
dbp:mode: cs1
dbo:abstract: Silnia liczby naturalnej n – iloczyn wszystkich liczb naturalnych dodatnich nie większych niż . Zapis itd. odczytujemy „n silnia”, „dwa silnia” itd. In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: For example,The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of distinct objects: there are . In mathematical analysis, factorials are used in power series for the exponential function and other functions, and they also have applications in algebra, number theory, probability theory, and computer science. Much of the mathematics of the factorial function was developed beginning in the late 18th and early 19th centuries.Stirling's approximation provides an accurate approximation to the factorial of large numbers, showing that it grows more quickly than exponential growth. Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials. Daniel Bernoulli and Leonhard Euler interpolated the factorial function to a continuous function of complex numbers, except at the negative integers, the (offset) gamma function. Many other notable functions and number sequences are closely related to the factorials, including the binomial coefficients, double factorials, falling factorials, primorials, and subfactorials. Implementations of the factorial function are commonly used as an example of different computer programming styles, and are included in scientific calculators and scientific computing software libraries. Although directly computing large factorials using the product formula or recurrence is not efficient, faster algorithms are known, matching to within a constant factor the time for fast multiplication algorithms for numbers with the same number of digits. Факториа́л — функция, определённая на множестве неотрицательных целых чисел. Название происходит от лат. factorialis — действующий, производящий, умножающий; обозначается , произносится эн факториа́л. Факториал натурального числа определяется как произведение всех натуральных чисел от 1 до включительно: . Например, . Для принимается в качестве соглашения, что . Факториал активно используется в различных разделах математики: комбинаторике, математическом анализе, теории чисел, функциональном анализе и др. Факториал является чрезвычайно быстро растущей функцией. Он растёт быстрее, чем любая показательная функция или любая степенная функция, а также быстрее, чем любая сумма произведений этих функций. Однако степенно-показательная функция растёт быстрее факториала, так же как и большинство двойных степенных, например . 在數學中，正整数的階乘（英語：Factorial）是所有小於等於該數的正整數的積，計為n!，例如5的階乘表示為5!，其值為120：並定義，1的階乘1!和0的階乘0!都為1，其中0的階乘表示一個空積。 1808年，基斯頓·卡曼引進這個表示法：，符號表示連續乘積，亦即n!=1×2×3×...×n。階乘亦可以遞迴方式定義：0!=1，n!=(n-1)!×n。除了自然數之外，階乘亦可定義于整個實數（負整數除外），其与伽瑪函數的关系为：階乘應用在許多數學領域中，最常應用在組合學、代數學和数学分析中。在組合學中，階乘代表的意義為n個相異物件任意排列的數量，例如前述例子，其代表了5個相異物件共有120種排列法。在正整數的情形下，n的階乘又可以稱為n的排列數。 En la matematiko, faktorialo de natura nombro n estas la produto de la pozitivaj entjeroj malpli aŭ egalaj al n. Oni signas ĝin per n!, kion oni prononcas no faktoriale laŭ Christian Kramp. Edozein n zenbakiaren faktoriala, n zenbaki arrunta izanik, 1 eta n artean dauden zenbaki natural guztien biderkaduraren emaitza da. Adibidez: n! notazioa matematikariak sortu zuen. In matematica, si definisce fattoriale di un numero naturale , indicato con , il prodotto dei numeri interi positivi minori o uguali a tale numero. In formula: per la convenzione del prodotto vuoto si definisce inoltre . La generalizzazione analitica del fattoriale è nota con il nome di funzione gamma di Eulero. La notazione con il punto esclamativo è stata introdotta nel 1807 da Christian Kramp, mentre il nome fattoriale era stato coniato pochi anni prima, nel 1800 da Antoine Arbogast. La sequenza dei fattoriali compare nella On-Line Encyclopedia of Integer Sequences (OEIS) come sequenza . El factorial de un entero positivo n, el factorial de n o n factorial se define en principio como el producto de todos los números enteros positivos desde 1 (es decir, los números naturales) hasta n. Por ejemplo: La operación de factorial aparece en muchas áreas de las matemáticas, particularmente en combinatoria y análisis matemático.De manera fundamental el factorial de n representa el número de formas distintas de ordenar n objetos distintos (elementos sin repetición). Este hecho ha sido conocido desde hace varios siglos, en el siglo XII por los estudiosos hindúes. La definición de la función factorial también se puede extender a números no naturales manteniendo sus propiedades fundamentales, pero se requieren matemáticas avanzadas, particularmente del análisis matemático. El matemático francés (1760-1826) fue la primera persona en usar la actual notación matemática n!, en 1808. De faculteit van een natuurlijk getal , genoteerd als (n faculteit), is het product van de getallen tot en met : Recursief geldt dus voor de faculteit: Voor bijvoorbeeld is: In overeenstemming met de definitie van het lege product is afgesproken dat De faculteitsfunctie groeit snel, zelfs sneller dan een exponentiële functie. De eerste 20 waarden, met nul, staan hiernaast. Het aantal decimalen van n! , met n > 1 , is gelijk aan 10log 1 + ... + 10log n naar boven afgerond. Voor n = 1000 komt het aantal decimalen op 2568. Факторіал натурального числа — добуток натуральних чисел від одиниці до включно, позначається !. . За означенням , згідно з конвенцією для .. При великих наближене значення факторіала можна обчислити за формулою Стірлінга. Факторіал дорівнює кількості перестановок з елементів. Στα μαθηματικά τo παραγοντικό ενός φυσικού αριθμού ν συμβολίζεται με ν!, διαβάζεται νι παραγοντικό, και είναι το γινόμενο όλων των θετικών ακεραίων μικρότερων ή ίσων με ν: ν! = 1 ∙ 2 ∙ 3 ∙ ... ∙ ν Για παράδειγμα, 2!=1·2= 2 3!=1·2·3= 6 4!=1·2·3·4= 24 5!=1·2·3·4·5= 120 8!=1·2·3·4·5·6·7·8= 40.320 10!=1·2·3·4·5·6·7·8·9·10= 3.628.800 12!=1·2·3·4·5·6·7·8·9·10·11·12= 479.001.600 Το παραγοντικό ενός αριθμού ν εκφράζει και το πλήθος των δυνατών μεταθέσεων των ν στοιχείων ενός συνόλου, δηλαδή το πλήθος των διαφορετικών τρόπων με τους οποίους μπορούμε να βάλουμε σε μια σειρά τα ν στοιχεία ενός συνόλου. * Συμβατικά: 0! = 1! = 1 * Ισχύει η σχέση: ν! = (ν-1)! ∙ ν Ένας άλλος τρόπος να προσεγγίσουμε το 0! είναι ακολουθόντας ένα μοτίβο το οποίο έχει ως εξής. 5!=120 4!=5!/5=24 3!=4!/4=6 2!=3!/3=2 1!=2!/2=1 0!=1!/1=1 V matematice je faktoriál čísla n (značeno pomocí vykřičníku: n!) číslo, rovné součinu všech kladných celých čísel menších nebo rovných n, pokud je n kladné, a rovno 1 pro n = 0. Značení n! vyslovujeme jako „n faktoriál“. Toto značení zavedl Christian Kramp v roce 1808. 数学において非負整数 n の階乗（かいじょう、英: factorial）n ! は、1 から n までの全ての整数の積である。例えば、である。空積の規約のもと 0! = 1 と定義する。階乗は数学の様々な場面に出現するが、特に組合せ論、代数学、解析学などが著しい。階乗の最も基本的な出自は n 個の相異なる対象を1列に並べる方法（対象の置換）の総数が n! 通りであるという事実である。階乗の定義は、最も重要な性質を残したまま、非整数を引数とする函数にすることができる。そうすれば解析学における著しい手法などの進んだ数学を利用できるようになる。 ( 계승(繼承)에 대해서는 왕위 계승 문서를 참고하십시오.) 수학에서, 자연수의 계승 또는 팩토리얼(階乘, 문화어: 차례곱, 영어: factorial)은 그 수보다 작거나 같은 모든 양의 정수의 곱이다. n이 하나의 자연수일 때, 1에서 n까지의 모든 자연수의 곱을 n에 상대하여 이르는 말이다. 기호는 을 쓰며 팩토리얼이라고 읽는다. 공식적이지는 않지만 한국 사람들 사이에서 팩토리얼을 줄여서 팩이라고 읽기도 한다. في الرياضيات، المضروب أو العاملي لعدد صحيح طبيعي n، والذي يكتب ، والذي يقرأ "عاملي n"، هو جداء كل الأعداد الطبيعية (الأعداد الصحيحة الموجبة قطعاً) المساوية أو الأصغر من n، ما عدا الصفر. فيما يلي مثال 5 عاملي: و تعريف العاملي على شكل جداء يترتب عنه كون ذلك أن 0! جداء مفرغ، وبمعنى آخر مختصر أي عدد مضروب في صفر يساوي صفر في عملية الضرب. تظهر دالة العاملي في مجالات مختلفة من الرياضيات، وخصوصا في التوافقيات والجبر والتحليل الرياضي. أبسط مثال على ذلك، وجود !n طريقة مختلفة لترتيب عناصر مجموعة عددهم مساو ل n (أي عدد التبديلات لعناصر هذه المجموعة). عرفت هذه الحقيقة على الأقل منذ القرن الثاني عشر الميلادي، من طرف علماء الرياضيات الهنديين. ويظهر العاملي في عدة معادلات رياضية، مثل صيغة الثنائي الحد لنيوتن وصيغة تايلور. إستُعمل رمز علامة التعجب (!) للتعبير عن دالة عاملي لأول مرة من طرف عالم الرياضيات كريستيان كرامب وكان ذلك عام 1808. يمكن لتعريف دالة عاملي أن يمدد إلى أعداد غير صحيحة بدون المساس بخصائص هذه الدالة. هذه العملية تستلزم تقنيات متطورة في الرياضيات وخصوصا تلك المستقاة من التحليل الرياضي. Na matemática, o fatorial (AO 1945: factorial) de um número natural n, representado por n!, é o produto de todos os inteiros positivos menores ou iguais a n. A notação n! foi introduzida por Christian Kramp em 1808. Die Fakultät (manchmal, besonders in Österreich, auch Faktorielle genannt) ist in der Mathematik eine Funktion, die einer natürlichen Zahl das Produkt aller natürlichen Zahlen (ohne Null) kleiner und gleich dieser Zahl zuordnet. Sie wird durch ein dem Argument nachgestelltes Ausrufezeichen („!“) abgekürzt. Diese Notation wurde erstmals 1808 von dem elsässischen Mathematiker Christian Kramp (1760–1826) verwendet, der um 1798 auch die Bezeichnung faculté „Fähigkeit“ dafür einführte. En matemàtiques, el factorial d'un enter no negatiu , denotat per (en alguns llibres antics es pot trobar denotat per ), és el producte de tots els nombres enters positius inferiors o iguals a . Per exemple, El valor de és 1, d'acord amb la convenció d'un producte buit. L'operació factorial es troba en moltes àrees de les matemàtiques, principalment en combinatòria, àlgebra i anàlisi matemàtica. La seva aparició més bàsica és el fet que hi ha formes d'organitzar objectes diferents en una seqüència (és a dir, permutacions del conjunt d'objectes). Aquest fet ja era conegut pels erudits indis, almenys ja al segle xii. En 1677, va descriure els factorials aplicats per . Després de descriure un enfocament recursiu, Stedman va donar una declaració de factorial (usant el llenguatge de l'original): La notació va ser introduïda pel matemàtic francès Christian Kramp el 1808. La definició de la funció factorial també es pot ampliar a arguments no enters, tot conservant les seves propietats més importants; això implica matemàtiques més avançades, especialment tècniques d'anàlisi matemàtica. En mathématiques, la factorielle d'un entier naturel n est le produit des nombres entiers strictement positifs inférieurs ou égaux à n. Cette opération est notée avec un point d'exclamation, n!, ce qui se lit soit « factorielle de n », soit « factorielle n », soit « n factorielle ». Cette notation a été introduite en 1808 par Christian Kramp. Par exemple, la factorielle 10 exprime le nombre de combinaisons possibles de placement des 10 convives autour d'une table (on dit la permutation des convives). Le premier convive s'installe sur l'une des 10 places à sa disposition. Chacun de ses 10 placements ouvre 9 nouvelles possibilités pour le deuxième convive, celles-ci 8 pour le troisième, et ainsi de suite. La factorielle joue un rôle important en algèbre combinatoire parce qu'il y a n! façons différentes de permuter n objets. Elle apparaît dans de nombreuses formules en mathématiques, comme la formule du binôme et la formule de Taylor. Dalam matematika, Faktorial dari bilangan bulat positif dari n yang dilambangkan dengan n!, adalah produk dari semua bilangan bulat positif yang kurang dari atau sama dengan n: Sebagai contoh, Nilai 0! adalah 1, menurut konvensi untuk . Operasi faktorial digunakan sebagai bidang matematika, terutama di kombinatorik, aljabar, dan analisis matematika. Penggunaannya yang paling dasar menghitung kemungkinan urutan dan permutasi dari n yang berada di objekk yang berbeda. Faktorial pada fungsi juga dapat berupa nilai ke argumen non-bilangan bulat sambil mempertahankan properti terpentingnya dengan cara mendefinisikan x! = Γ(x + 1), di mana Γ adalah fungsi gamma; ini tidak ditentukan saat x adalah bilangan bulat negatif. Fakultet är en funktion inom matematiken. För ett heltal större än noll är fakulteten lika med produkten av alla heltal från 1 upp till och med talet självt.
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