Trinomial expansion

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

${\displaystyle {\begin{aligned}(a+b+c)^{n}&=(a+d)^{n}=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,d^{r}\\&=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,(b+c)^{r}\\&=\sum _{r=0}^{n}{n \choose r}\,a^{n-r}\,\sum _{s=0}^{r}{r \choose s}\,b^{r-s}\,c^{s}.\end{aligned}}}$

Above, the resulting ${\displaystyle (b+c)^{r}}$ in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index ${\displaystyle s}$.

The product of the two binomial coefficients is simplified by shortening ${\displaystyle r!}$,

${\displaystyle {n \choose r}\,{r \choose s}={\frac {n!}{r!(n-r)!}}{\frac {r!}{s!(r-s)!}}={\frac {n!}{(n-r)!(r-s)!s!}},}$

and comparing the index combinations here with the ones in the exponents, they can be relabelled to ${\displaystyle i=n-r,~j=r-s,~k=s}$, which provides the expression given in the first paragraph.

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

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

where n is the exponent to which the trinomial is raised.^[3]

An example of a trinomial expansion with ${\displaystyle n=2}$ is :

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

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