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 – 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
:<math>(a+b+c)^n = \sum_ {n \choose i,j,k}\, a^i \, b^{\;\! j} \;\! c^k, </math><!-- \;\! yields +5 -3 = 2mu space -->
where is a nonnegative integer and the sum is taken over all combinations of nonnegative indices and such that . 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 . The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron.
Derivation
The trinomial expansion can be calculated by applying the 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 is the exponent to which the trinomial is raised.
Example
Examples of trinomial expansions with <math>n=2,3,4</math> are:
<math>(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ac)</math>
<math>(a+b+c)^3=a^3+b^3+c^3+3(a^2b+ab^2+b^2c+bc^2+ac^2+a^2c) + 6abc</math>
<math>(a+b+c)^4=a^4+b^4+c^4+4(a^3b+ab^3+b^3c+bc^3+a^3c+ac^3) + 6(a^2b^2+b^2c^2+a^2c^2) + 12(a^2bc+ab^2c+abc^2)</math>
See also
- Binomial expansion
- Pascal's pyramid
- Multinomial coefficient
- Trinomial triangle