I am taking a course in PDE and trying to get use to these notations. If I take
3 vectors with nonnegative integers components (n components) that are denoted by $\alpha$,$\omega$,$\gamma$. Is it correct to say that Pascal's formula can be written in the following form?
\begin{equation} \left(\begin{array}{c} \alpha-\gamma \\ \omega-\gamma \end{array}\right)+\left(\begin{array}{c} \alpha-\gamma \\ \omega \end{array}\right)=\left(\begin{array}{c} \alpha \\ \omega \end{array}\right) \end{equation}
where \begin{equation} \left(\begin{array}{l} \alpha \\ \beta \end{array}\right)=\frac{\alpha !}{\beta !(\alpha-\beta) !} \end{equation}
and \begin{equation} \boldsymbol{\alpha} !=\alpha_{1} ! \ldots \alpha_{n} ! \end{equation}
I have tried to prove that but I can't see connection between the numerator and denominator in each term (L.H.S).