Trinomial expansion
{{Short description|Formula in mathematics}}
File:Pascal_pyramid_trinomial.svg derived from coefficients in an upside-down ternary plot of the terms in the expansions of the powers of a trinomial – {{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 monomials. The expansion is given by
:
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}}.{{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}}. The trinomial coefficients are given by
:
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.{{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}}.
Derivation
The trinomial expansion can be calculated by applying the binomial expansion twice, setting , which leads to
:
\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}
Above, the resulting in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index .
The product of the two binomial coefficients is simplified by shortening ,
:
{n \choose r}\,{r \choose s} = \frac{n!}{r!(n-r)!} \frac{r!}{s!(r-s)!}
= \frac{n!}{(n-r)!(r-s)!s!},
and comparing the index combinations here with the ones in the exponents, they can be relabelled to , which provides the expression given in the first paragraph.
Properties
The number of terms of an expanded trinomial is the triangular number
:
where {{math|n}} is the exponent to which the trinomial is raised.{{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 }}.
Example
An example of a trinomial expansion with is :