Pascal's rule : I don't understand why we can do this... So I understand that they are equal, but I can't get my head around the wording.  Why can we choose "1 special element, and $k − 1$ from the remaining $n − 1$, or choose all $k$ from the $n − 1$"?

To choose $k$ elements from $n$ we can either choose 1 special element, and $k-1$ from the remaining $n-1$, or choose all $k$ from the $n-1$.  This [sic] we have Pascal's Rule $$C_k^n = C_{k-1}^{n-1} + C_k^{n-1}. $$ This leads on to the so-called Pascal's Triangle.

Sorry if this basic but sometimes I have an issue getting around wording I'm hope there is a simpler wording for it!

It just explains that, once you've marked one (fixed) element, choosing $k$ elements from a set of $n$ can be done in two (incompatible) ways:


*

*either this marked element is in your choice, and there are $\dbinom{n-1}{k-1}$ ways to choose the $k-1$ other among the remaining $n-1$;

*or it is not, and you have $\dbinom{n-1}k$ possible ways.

