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!

To see the correspondence, note that the number $C_k^n = \binom{n}{k}$ is the number of a teams we can construct of size $k$ with $n$ players. Now assign one of the players as "special". This player can be either included or not included in the team, so if it's included there are $C_{k-1}^{n-1} = \binom{n-1}{k-1}$ ways to choose the rest players. If he's not we can choose the team in $C_k^{n-1} = \binom{n-1}{k}$ ways.
The punchline is that the special player must be included OR not included in the team and they are two disjoint cases, so we can find the number of combinations in both cases and add them to obtain the total sum.
