The problem is to come up with story proof for the following:

Show that for all positive integers n and k with n $\geq$ k, $$\mathbf{\binom{n}{k}+ \binom{n}{k-1} = \binom{n+1}{k}},$$ doing this with a story, giving an interpretation for why both sides count the same thing.
Hint for the story proof: Imagine n + 1 people, with one of them pre-designated as “president”.

The [explanation](https://premmi.github.io/probability-story-proofs-part-I#:%7E:text=Suppose%20from%20a%20group,LHS%7D%24%20of%20the%20identity.) I found is following:

> 
> Suppose from a group of $\mathbf{n+1}$ people we have to form a
> committee comprising of $\mathbf{k}$ people. Also in this group one
> person is pre-designated as president. The number of ways of forming
> this committee is given by $\mathbf{\binom{n+1}{k}}$ which is the
> **right-hand side** of the identity. We can also think of forming the committee in two ways, one in which the president $\mathbf{is}$ chosen
> and out of the remaining $\mathbf{n}$ people we choose $\mathbf{k-1}$
> and one in which the president is $\mathbf{not}$ chosen and we choose
> $\mathbf{k}$ people from $\mathbf{n}$ people. Summing these two cases
> we get $\mathbf{\binom{n}{k-1} + \binom{n}{k}}$ which is the
> **left-hand side** of the identity.
> 
> 
> 

I have no questions regarding the second term "*n choose k*" — the (n+1)th person that was left out is the pre-designated as president.
However the first term baffles me. If the (n+1)th person isn't a president than how the term can account for all group, and how the committee is guaranteed to include the pre-designated president.

I know that I probably don't understand something obvious and would be grateful if someone could point me in the right direction.