Template:Binomial

The following documentation is located at Template:Binomial/doc. [<Edit> Template:Binomial/doc]
[⧼Purge⧽ Template:Binomial]

The {{binomial}}, {{binom}} or {{C(n,k)}} mathematical formatting template and/or mathematical function template returns either the typeset (HTML+CSS or LaTeX) expression of the the binomial coefficient or the numerical result, for nonnegative integers

n

and

k

, of

(  nk  )

.

Usage

{{binomial|nonnegative integer n|nonnegative integer k|format}}

or

{{binom|nonnegative integer n|nonnegative integer k|format}}

or

{{C(n,k)|nonnegative integer n|nonnegative integer k|format}}

where format is

• htm: HTML+CSS text style (default), or
• HTM: HTML+CSS display style, or
• tex: LaTeX text style, or
• TEX: LaTeX display style, or
• #: numerical result.

Examples

The code

: before {{math|{{binom|''n''|''k''|htm}}|&}} after

yields the text style HTML+CSS

before 

(  nk  )

after

The code

: before {{math|{{binom|''n''|''k''|tex}}|$}} after

yields the text style LaTeX

before$\textstyle {{\binom {n}{k}}\!}$ after

The code

{{indent}}{{math|{{binom|''n''|''k''|HTM}}|&&}}

yields the display style HTML+CSS

     

( n k )

The code

: {{math|{{binom|''n''|''k''|TEX}}|$$}}

yields the display style LaTeX

${\begin{array}{l}\displaystyle {{\binom {n}{k}}\!}\end{array}}$

The code

: before {{math|{{binom|9|3|htm}} {{=}} {{binom|9|3|#}}|&}} after

yields the text style HTML+CSS

before 

(  93  )  = 84

after

The code

: before {{math|{{binom|9|3|tex}} {{=}} {{binom|9|3|#}}|$}} after

yields the text style LaTeX

See Template:Binomial/doc (transcluding the /doc subpage is one transclusion too many)

Examples with numerical result

The algorithm for the evaluation of binomial coefficients uses

     

( n k )  :=  n (n  − k ) k !  =  ∏ n  − k  − 1 i  = 0  (n − i ) k ! , n ≥ 0, k ≥ 0,

where

n (m)   :=   ∏ m  − 1 i  = 0  (n  −  i )

is the falling factorial.

Examples with numerical result: valid input

Numerical tests.— Daniel Forgues 21:55, 11 March 2018 (EDT)

Code	Result	Comment	Code	Result	Comment
{{binomial|0|0|#}}	1	Y	{{binom|0|1|#}}	0	Y
{{binomial|1|1|#}}	1	Y	{{binom|2|0|#}}	1	Y
{{binomial|2|1|#}}	2	Y	{{binom|2|2|#}}	1	Y
{{C(n,k)|3|1|#}}	3	Y	{{C(n,k)|4|2|#}}	6	Y
{{C(n,k)|5|3|#}}	10	Y	{{C(n,k)|6|2|#}}	15	Y
{{C(n,k)|7|3|#}}	35	Y	{{C(n,k)|8|3|#}}	56	Y
{{C(n,k)|9|3|#}}	84	Y	{{C(n,k)|10|3|#}}	120	Y
{{C(n,k)|11|4|#}}	330	Y	{{C(n,k)|12|4|#}}	495	Y
{{C(n,k)|13|5|#}}	1287	Y	{{C(n,k)|14|5|#}}	2002	Y
{{C(n,k)|15|3|#}}	455	Y	{{C(n,k)|16|3|#}}	560	Y
{{C(n,k)|17|3|#}}	680	Y	{{C(n,k)|17|23|#}}	0	Y
{{C(n,k)|30|21|#}}	14307150	Y	{{C(n,k)|47|13|#}}	140676848445	Y
{{C(n,k)|48|19|#}}	11541847896480	Y	{{C(n,k)|49|19|#}}	18851684897585	Y (  4919  )  = 18851684897584 (rounding error)*
{{C(n,k)|80|3|#}}	82160	Y	{{C(n,k)|83|3|#}}	91881	Y
{{C(n,k)|99|31|#}}	4.5764000431736E+25	Y	{{C(n,k)|91|3|#}}	121485	Y

* 

(  4919  )  = 49(30) / 19! = 2293215102242267478449061888000 / 121645100408832000 = 18851684897584

(rounding error).

Examples with numerical result: invalid input

Code	Result
{{C(n,k)|-1|3|#}}
{{C(n,k)|0.5|3|#}}
{{C(n,k)|text|3|#}}
{{C(n,k)|6 blobs|3|#}}
{{C(n,k)|12|nine|#}}
{{C(n,k)|99|43|#}}
{{C(n,k)|99|33|#}}

See also

• {{factorial}} or {{n!}}
• {{Pochhammer}} or {{Poch}} or {{x^(n)}}
• {{binomial}} or {{binom}} or {{C(n,k)}}
• {{atop}}