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


 * $$(a+b+c)^n = \sum_{{i,j,k}\atop{i+j+k=n}} {n \choose i,j,k}\, a^i \, b^{\;\! j} \;\! c^k, $$

where $n$ is a nonnegative integer and the sum is taken over all combinations of nonnegative indices $i, j,$ and $k$ such that $i + j + k = n$. The trinomial coefficients are given by


 * $$ {n \choose i,j,k} = \frac{n!}{i!\,j!\,k!} \,.$$

This formula is a special case of the multinomial formula for $m = 3$. The coefficients can be defined with a generalization of Pascal's triangle to three dimensions, called Pascal's pyramid or Pascal's tetrahedron.

Derivation
The trinomial expansion can be calculated by applying the binomial expansion twice, setting $$d = b+c$$, 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 $$(b+c)^{r}$$ in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index $$s$$.

The product of the two binomial coefficients is simplified by shortening $$r!$$,

{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 $$i=n-r, ~ j=r-s, ~ k = s$$, which provides the expression given in the first paragraph.

Properties
The number of terms of an expanded trinomial is the triangular number


 * $$ t_{n+1} = \frac{(n+2)(n+1)}{2}, $$

where $n$ is the exponent to which the trinomial is raised.

Example
An example of a trinomial expansion with $$n=2$$ is :

$$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$$