Editing Trinomial expansion

{{Short description|Formula in mathematics}}
[[File:Pascal_pyramid_trinomial.svg|thumb|upright=2|Layers of [[Pascal's pyramid]] derived from coefficients in an upside-down [[ternary plot]] of the terms in the expansions of the powers of a trinomial &ndash; {{nowrap|the number of terms}} is clearly a [[triangular number]] ]]
In [[mathematics]], a '''trinomial expansion''' is the expansion of a power of a sum of three terms into [[monomial]]s. The expansion is given by

:<math>(a+b+c)^n = \sum_{{i,j,k}\atop{i+j+k=n}}  {n \choose i,j,k}\, a^i \, b^{\;\! j} \;\! c^k, </math><!-- \;\! yields +5 -3 = 2mu space -->

where {{math|''n''}} is a nonnegative integer and the sum is taken over all combinations of nonnegative indices {{math|''i'', ''j'',}} and {{math|''k''}} such that {{math|''i'' + ''j'' + ''k'' {{=}} ''n''}}.<ref>{{citation|title=Discrete Mathematics with Applications|first=Thomas|last=Koshy|publisher=Academic Press|year=2004|isbn=9780080477343|url=https://books.google.com/books?id=90KApidK5NwC&pg=PA889|page=889}}.</ref> The '''trinomial coefficients''' are given by

:<math> {n \choose i,j,k} = \frac{n!}{i!\,j!\,k!} \,.</math>

This formula is a special case of the [[multinomial formula]] for {{math|''m'' {{=}} 3}}. The coefficients can be defined with a generalization of [[Pascal's triangle]] to three dimensions, called [[Pascal's pyramid]] or Pascal's tetrahedron.<ref>{{citation|title=Combinatorics and Graph Theory|series=[[Undergraduate Texts in Mathematics]]|first1=John|last1=Harris|first2=Jeffry L.|last2=Hirst|first3=Michael|last3=Mossinghoff|edition=2nd|publisher=Springer|year=2009|isbn=9780387797113|page=146|url=https://books.google.com/books?id=DfcQaZKUVLwC&pg=PA146}}.</ref>

==Derivation==
The trinomial expansion can be calculated by applying the [[Binomial theorem|binomial expansion]] twice, setting <math>d = b+c</math>, which leads to

:<math>
\begin{align}
(a+b+c)^n &= (a+d)^n = \sum_{r=0}^{n} {n \choose r}\, a^{n-r}\, d^{r} \\
	&= \sum_{r=0}^{n} {n \choose r}\, a^{n-r}\, (b+c)^{r} \\
	&= \sum_{r=0}^{n} {n \choose r}\, a^{n-r}\, \sum_{s=0}^{r} {r \choose s}\, b^{r-s}\,c^{s}.
\end{align}
</math>

Above, the resulting <math>(b+c)^{r}</math> in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index <math>s</math>.

The product of the two binomial coefficients is simplified by shortening <math>r!</math>,
:<math>
{n \choose r}\,{r \choose s} = \frac{n!}{r!(n-r)!} \frac{r!}{s!(r-s)!}
= \frac{n!}{(n-r)!(r-s)!s!},
</math>

and comparing the index combinations here with the ones in the exponents, they can be relabelled to <math>i=n-r, ~ j=r-s, ~ k = s</math>, which provides the expression given in the first paragraph.

== Properties ==
The number of terms of an expanded trinomial is the [[triangular number]]

:<math> t_{n+1} = \frac{(n+2)(n+1)}{2}, </math>

where {{math|''n''}} is the exponent to which the trinomial is raised.<ref>{{citation|last=Rosenthal|first=E. R.|title=A Pascal pyramid for trinomial coefficients|journal=The Mathematics Teacher|year=1961|volume=54|issue=5|pages=336–338|doi=10.5951/MT.54.5.0336 }}.</ref>

== Example ==
An example of a trinomial expansion with <math>n=2</math> is :

<math>(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca</math>

==See also==
* [[Binomial expansion]]
* [[Pascal's pyramid]]
* [[Multinomial coefficient]]
* [[Trinomial triangle]]

==References==
{{reflist}}

{{DEFAULTSORT:Trinomial Expansion}}
[[Category:Factorial and binomial topics]]