Trinomial expansion

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 – Template:Nowrap 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

$(a+b+c{)}^n=\sum _{\genfrac{}{}{0ex}{}{i,j,k}{i+j+k=n}}(\genfrac{}{}{0ex}{}{n}{i,j,k})a^i{b}^j{c}^k,$

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. The trinomial coefficients are given by

$(\genfrac{}{}{0ex}{}{n}{i,j,k})=\frac{n!}{i!j!k!}.$

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.

Derivation

The trinomial expansion can be calculated by applying the binomial expansion twice, setting $d=b+c$, which leads to

$\begin{array}{cc}(a+b+c{)}^n& =(a+d{)}^n=\sum _{r=0}^n(\genfrac{}{}{0ex}{}{n}{r})a^{n-r}{d}^r\\ & =\sum _{r=0}^n(\genfrac{}{}{0ex}{}{n}{r})a^{n-r}(b+c{)}^r\\ & =\sum _{r=0}^n(\genfrac{}{}{0ex}{}{n}{r})a^{n-r}\sum _{s=0}^r(\genfrac{}{}{0ex}{}{r}{s})b^{r-s}{c}^s.\end{array}$

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 $r!$,

$(\genfrac{}{}{0ex}{}{n}{r})(\genfrac{}{}{0ex}{}{r}{s})=\frac{n!}{r!(n-r)!}\frac{r!}{s!(r-s)!}=\frac{n!}{(n-r)!(r-s)!s!},$

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 Template:Math is the exponent to which the trinomial is raised.

Example

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

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

See also

• Binomial expansion
• Pascal's pyramid
• Multinomial coefficient
• Trinomial triangle