<s>[INST] Proportion with 3 variables - binomial coefficients So, if we're given something like this:
$$\binom{n}{k}:\binom{n+1}{k}:\binom{n+1}{k+1}=3:4:8$$
How do I rewrite this so I can manipulate it?
Edit: Is there a general procedure for n variables?
 [/INST] 
Note, that the equality of proportions
  \begin{align*}
a:b:c=x:y:z
\end{align*}
  is a compact notation for
  \begin{align*}
\frac{a}{b}=\frac{x}{y},\qquad\frac{a}{c}=\frac{x}{z},\qquad\frac{b}{c}=\frac{y}{z}\tag{1}
\end{align*}
  Each equality in (1) can be derived from the other two.

So, in order to check if we can solve
\begin{align*}
\binom{n}{k}:\binom{n+1}{k}:\binom{n+1}{k+1}=3:4:8
\end{align*}
we take two proportions and calculate $n$ and $k$.

We find by solving
  \begin{align*}
\frac{\binom{n}{k}}{\binom{n+1}{k}}=\frac{3}{4}\qquad \frac{\binom{n+1}{k}}{\binom{n+1}{k+1}}=\frac{4}{8}
\end{align*}
the solutions $n=7,k=2$ and can finally conclude
  \begin{align*}
\binom{7}{2}:\binom{8}{2}:\binom{8}{3}=21:28:56=3:4:8
\end{align*}

</s>