Trinomial expansion
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_{{i,j,k}\atop{i+j+k=n}} {n \choose i,j,k}\, a^i \, b^{\;\! j} \;\! c^k, </math>
where Template:Math is a nonnegative integer and the sum is taken over all combinations of nonnegative indices Template:Math and Template:Math such that Template:Math.<ref>Template:Citation.</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 Template:Math. The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron.<ref>Template:Citation.</ref>
DerivationEdit
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.
PropertiesEdit
The number of terms of an expanded trinomial is the triangular number
- <math> t_{n+1} = \frac{(n+2)(n+1)}{2}, </math>
where Template:Math is the exponent to which the trinomial is raised.<ref>Template:Citation.</ref>
ExampleEdit
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>