Discrete Math - Combinatorics - Trinomial Coefficients question Let $k,l,m,n \in Z \geq 0$ be such that $n=k+l+m$. The trinomial coefficient ${n \choose k,l,m}$ is given by the rules:


*

*for $k+l=n$, ${n\choose k,l,0} = {n \choose k,0,l} = {n \choose 0,k,l} = {n\choose k}$

*${n\choose k,l,m} = {n-1 \choose k-1,l,m} + {n-1 \choose k,l-1,m} + {n-1\choose k,l,m-1}$
The following questions use this definition.
(a) What are all the trinomial coefficients for $n=1,2,3$?
(b) Describe the "triangle" of trinomial coefficients (Hint: Think three dimensional Pascal's triangle). 
I don't understand the notation at all: n choose k,l,m. What does that even mean? And I don't get how to set up part (a) at all either. It's just very confusing. For (b) I can visualize a Pascal's "pyramid." One where the number below is a sum of the 3 above it. Something like that. But other than that, I'm not really sure what's going on.
 A: The lines numbered (1) and (2) are the definition of $\binom{n}{k,\ell,m}$, so it means exactly what they say it means. It’s a recursive definition: it tells you how to compute trinomial coefficients with upper number $n$ if you already know how to compute them with upper number $n-1$. However, you’ve miscopied (2), unless there was a typo in your source: it should read
$$\binom{n}{k,\ell,m}=\binom{n-1}{k-1,\ell,m}+\binom{n-1}{k,\ell-1,m}+\binom{n-1}{k,\ell,m-1}\;.$$
Here’s an example to illustrate how to use (1) and (2) to calculate a trinomial coefficient:
$$\begin{align*}
\binom4{1,2,1}&=\binom3{0,2,1}+\binom3{1,1,1}+\binom3{1,2,0}&\text{using (2)}\\\\
&=\binom32+\binom3{1,1,1}+\binom31&\text{using (1)}\\\\
&=\binom32+\binom2{0,1,1}+\binom2{1,0,1}+\binom2{1,1,0}+\binom31&\text{using (2)}\\\\
&=\binom32+\binom21+\binom21+\binom21+\binom31&\text{using (1)}\\\\
&=3+2+2+2+3\\
&=12\;.
\end{align*}$$
The pyramid that you describe is exactly what’s wanted for (b). At the peak you have $\binom0{0,0,0}$, which is $1$. Below that you have a triangle of three $1$’s. The next layer down will have six entries forming a triangle, corresponding to $\binom1{1,0,0},\binom1{0,1,0}$, and $\binom1{0,0,1}$. The next layer will have ten entries in a triangle, corresponding to the ten possible trinomial coefficients $\binom2{k,\ell,m}$ with $0\le k,\ell,m$ and $k+\ell+m=2$. And so on.
