Talk:Trinomial expansion

Ambiguous terminology

This article claims "trinomial coefficient" is used to designate a particular kind of multinomial coefficient, namely a coefficient of $(X+Y+Z)^{n}$, which would seem natural. However, I think that the term is actually more used for the largely unrelated notion of a coefficient of $(1+X+X^{2})^{n}$, see for instance this MathWorld article. So this article should either acknowledge the ambiguity, or adapt to that other usage altogether (I think it can easily be sourced with published references, which I doubt for the usage in the sense of the current article). Marc van Leeuwen (talk) 12:36, 25 June 2010 (UTC)

Although I am an experienced mathematician I haven't ever seen Wolfram's definition of "trinomial coefficient"; I've only ever seen "trinomial coefficient" to indicate a particular kind of multinomial coefficient. Let's go the route of acknowledging the ambiguity. Quantling (talk) 17:19, 25 June 2010 (UTC)

coefficient expressed as product of two binomial coefficients

In the article Pascal's triangle at the section on the relationship with Pascal's triangle, the fact that each "trinomial" coefficient is a product of two binomial coefficients and its geometric representation are discussed.

I would find it instructive to add something like this also to this article, where the expansion is relabelled to read the following:

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

$=\sum _{j=0}^{n}{n \choose j}\,a^{n-j}\sum _{k=0}^{j}{j \choose k}\,b^{j-k}c^{k}.$

The advantage of this representation is the fact that it already contains the previously given constriction of $i+j+k=n$. Unfortunately, this cannot be translated directly to the line below, because the meaning of the summation indices changes between the already displayed step and the step which I am proposing here. However, in the article linked above, the come up with this symbolic expression

$C(i,j)\times C(n,i)=C(n,i,j),\quad 0\leq i\leq n,\ 0\leq j\leq i,$

which immediately followed from the depicted symmetry of the tetrahedron. With a minimal description, these arrays could also be reduced to less screen height, when displayed as

${\begin{matrix}{\text{ 1}}\\{\text{ 5}}\quad {\text{ 5}}\\{\text{ 10}}\quad {\text{ 20}}\quad {\text{ 10}}\\{\text{ 10}}\quad {\text{ 30}}\quad {\text{ 30}}\quad {\text{ 10}}\\{\text{ 5}}\quad {\text{ 20}}\quad {\text{ 30}}\quad {\text{ 20}}\quad {\text{ 10}}\\{\text{ 1}}\quad {\text{ 5}}\quad {\text{ 10}}\quad {\text{ 10}}\quad {\text{ 5}}\quad {\text{ 1}}\end{matrix}}={\begin{matrix}1\\5\\10\\10\\5\\1\end{matrix}}\times {\begin{matrix}{\text{ 1}}\\{\text{ 1}}\quad {\text{ 1}}\\{\text{ 1}}\quad {\text{ 2}}\quad {\text{ 1}}\\{\text{ 1}}\quad {\text{ 3}}\quad {\text{ 3}}\quad {\text{ 1}}\\{\text{ 1}}\quad {\text{ 4}}\quad {\text{ 6}}\quad {\text{ 4}}\quad {\text{ 1}}\\{\text{ 1}}\quad {\text{ 5}}\quad {\text{ 10}}\quad {\text{ 10}}\quad {\text{ 5}}\quad {\text{ 1}}\end{matrix}}$

Someone with a good taste in design might still improve on this and additionally fix the spacing of the ones in the lower left of each triangle. I copied and adapted this from Pascal's_triangle#Number_of_elements_of_simplices.

P.S. For this topic, I did not bother to see how multi-line equations work in the wiki-math, but only added an additional line of <math>

Regards, Anoon Bondara (talk) 23:57, 22 February 2023 (UTC)