How do I do combinatoric algebra without tedious factorial multiplication and division?

I find combinatoric algebra very non-intuitive. I'm talking about Pascal's Identity $n\geq r$,
$$
\binom{n+1}{r}=\binom{n}{r}+\binom{n}{r-1}.
$$

I understand the tedious proof of the theorem but what's a trick for understanding combinatoric algebra in general? I can't eyeball and decompose a binomial without memorizing the formulas or doing tedious factorial multiplication and division.

It's never obvious how combinatoric algebra works:

EXample: