1805 C. Buchanan Mem. Expediency Eccl. Establishm. Brit. India i. i. p. i The present establishment of English chaplains for the British empire in India, is not much greater than the factorial establishment in the time of Lord Clive.

1885 Science 6 100 The advisability of..securing a limited district for a..‘factorial establishment’ for American citizens in that region [sc. the Kongo country].

1902 Jrnl. Soc. Arts 14 Mar. 345/2 Go through every detail of the factory work, including the management, both factorial and commercial.

1925 Sask. Co-operative News Apr. 8/2 The growth of factorial production by intricate, highly specialized machinery has left agriculture dependent on the manufacturers for much that they would formerly have produced for themselves.

1999 Q. Y. Yu Implementation China's Sci. & Technol. Policy xii. 111 The standardized, factorial and commercialized construction of houses for peasants.

1805 College of Fort William Bengal viii. 164 As the Writers generally arrive in India at the early age of sixteen and seventeen; it sometimes happens that..so far from knowing any thing of the rudiments of science,..they are strangers even to factorial arithmetic.

1816 J. F. W. Herschel in tr. S. F. Lacroix Elem. Treat. Differential & Integral Calculus 469 If u_x be the general term of an arithmetical progression, or if u_x = a + bx, the factor u_{x + n + 1} − u_x becomes (n + 1)b, which being constant, it appears that the difference of a factorial function of this kind is of the same form with the function itself, only having one factor less.

1834 Rep. Brit. Assoc. Advancem. Sci. 1833 222 Legendre has named the function Γ(1 + r) = 1.2…r, the function gamma. Kramp, who has written largely upon its properties, gave it..the name of faculté numérique; but in his subsequent memoirs..he has adopted the name of factorial function, which Arbagost [sic] proposed, and which I think it expedient to retain.

1860 G. Boole Calculus of Finite Differences 6 The term in which they are involved is called a factorial term.

1879 Amer. Jrnl. Math. 2 141 The expression x^(m) may be reduced by a suitable alteration of the variable, to a factorial form.

1920 Bull. Amer. Math. Soc. 26 128 The theory of the divisibility of factorial expressions seems to have been studied for the first time by Leibniz.

1956 Canad. Jrnl. Math. 8 243 The computation of moments is most easily carried out by the use of the notion of a factorial moment.

2007 S. T. Karris Numerical Anal. (ed. 3) vii. 40 Factorial polynomials provide an easier method of constructing a difference table.

1820 Ladies' Lit. Cabinet 21 Oct. 186/3 They had..a factorial meeting to affix the shipping prices for 1819.

1836 Cases Court of Session 14 364 The factorial management of the trust fell entirely into the hands of Paterson.

1881 J. S. Blackie Lay Serm. v. 177 Large farms and factorial management have formed together an unholy alliance.

1907 Sc. Law Reporter 44 556/1 Before the said farms..became vacant the pursuer had carried on..his ordinary factorial duties in terms of said engagement as factor.

1995 D. Hewitt in W. Scott Antiquary 517 Man of business, lawyer who..conducted routine legal and factorial work for the client.

1902 Wilson's Photogr. Mag. Jan. 3/1 It was in 1894 that Mr. Watkins first brought out his system of factorial development, and it is worthy of note that previous to that date he had regularly practised tentative development with pyro-ammonia.

1902 A. Watkins Watkins Man. 74 The plan was named by me the Eikronometer method; it has also been termed the Factorial method.

1922 Photo-era Mag. July 35/1 Factorial development, however, tends to produce over-contrast in underexposed negatives, and under-contrast in overexposed negatives.

1970 Focal Encycl. Photogr. (Desk ed.) 424/2 Factorial development gives correct results only with correctly exposed negatives.

2003 R. W. Lambrecht & C. Woodhouse Way beyond Monochrome 29/5 It is best to develop filter-base papers using factorial development.

1907 Lancet 19 Jan. 185/2 In treating of the mortality of London in 1905.., Sir Shirley Murphy reviews the ordinary methods of correcting death-rates for differences in the age and sex constitution of populations. These methods he illustrates by reference more particularly to two—the one generally known as the factorial method, the other, and more accurate, consisting in the application of the deaths at each age to a population of standard composition.

1935 R. A. Fisher Design Exper. vi. 111 With factorial experiments designed to make a large number of comparisons there will..usually be an ample number of degrees of freedom for the estimation of error.

1950 W. G. Cochran & G. M. Cox Exper. Designs v. 124 If all factors are independent in their effects, the factorial approach will result in a considerable saving of the time and material devoted to the experiments.

2000 New Scientist 2 Sept. 89/1 (advt.) EDIT is a six year, nine-centre, double blind, factorial design study of 631 subjects at risk of diabetes.

1813 Mem. Analyt. Soc. p. xiv In analytical investigations, we frequently meet with a series of quantities connected together by multiplication, whose differences are constant. These have received various names from different Geometers; Vandermonde called them powers of the second order; Kramp adopted the appellation of Facultes numeriques; and Arbogast named them Factorials.

1816 J. F. W. Herschel in tr. S. F. Lacroix Elem. Treat. Differential & Integral Calculus App. §370 Any quantity of the form axⁿ + bx^n—1 + &c. may be resolved into factorials.

1860 G. Boole Treat. Calculus Finite Differences ii. 6 In the expression sin x sin (x + h) sin (x + 2h)…sin {x + (m − 1)h}, the factors are usually called factorials, and the term in which they are involved is called a factorial term.

1879 Amer. Jrnl. Math. 2 350 n^{r )} represents the factorial n(n − 1)(n − 1)…(n − r + 1).

1907 Proc. Edinb. Math. Soc. 1906–7 25 119 The product of two factorials (x + a + 1)…(x + a + s) and (x + b + 1)…(x + b + s) is not altered by the substitution of −x −a −b −s −1 for x.

1979 E. S. Page & L. B. Wilson Introd. Computational Combinatorics ii. 14 At this stage the powers in the polynomial in n should be replaced by the descending factorials.

2004 S. B. Maurer & A. Ralston Discrete Algorithmic Math. (ed. 3) v. 497 These are the falling factorials... n_(k) = n(n − 1)(n − 2)(n − k + 1).

1869 J. J. Raven Church Bells Cambridgeshire 38 The number of changes that can be rung on a peal of bells is the factorial of the number of the bells.

1936 E. J. Berg Heaviside's Operational Calculus (ed. 2) xxiii. 118 The so-called π function (pi function) is the generalized factorial. Its value can be obtained from the gamma function or directly by series expansion.

1962 Arithm. Teacher 9 308/2 Four factorial (4!) means 4×3×2×1 or 24.

1993 E. S. Raymond New Hacker's Dict. (ed. 2) 62 Computer languages that provide bignums can perform exact calculations on very large numbers, such as 1000! (the factorial of 1000, which is 1000 times 999 times 998 times…times 2 times 1).

1998 S. Budiansky If Lion could Talk v. 117 Finding the most efficient order in which to visit a series of sites is known as the traveling salesman problem... The number of alternatives grows as the factorial of the number of sites to be visited.

2012 M. A. Freitag Math. Elem. School Teachers xv. 968 The number of permutations of n objects taken r at a time can also be written in terms of factorials involving n and r.

1845 T. Tate (title) A treatise on factorial analysis with the summation of series; containing various new developments of functions, &c.

1909 W. Bateson Mendel's Princ. Heredity (new ed.) xiv. 256 The question may be asked whether..any large classes of phenomena are encountered which are altogether inconsistent with the Mendelian system to such a degree as to suggest that they are quite incompatible with any scheme of factorial analysis.

1935 Human Factor 9 182 With this infinite number of possible factorial analyses of any set of correlations,..a psychologist can find one which agrees with any preconceived notions he may have about the linkages between tests.

2012 U. Klein et al. in M. Lamers & P. de Swart Case, Word Order & Prominence 62 The results were analyzed by means of a factorial analysis of variance (by subjects and by items).