When n and k are positive integers.
Would this be a correct proof?
Description: Suppose how many ways can we choose k people from a group of n + 1.
LHS: The left-hand side gives us the number of ways to do this by definition, on how to select k from n+1.
RHS: The right-hand side gives us the number of ways to do this as well, but instead, it first counts the number of elements plus one (n+1) times the number of ways to choose an element a subset of k - 1 in n over the number of elements k.
It suffices to provide a combinatorial proof of
$$ \binom{k}{1} \binom{n+1}{k} = \binom{n+1}{1} \binom{n}{k-1}$$.
Indeed, since $\binom{\ell}{1} = \ell$, we can apply this to $\ell = n,k$ and then move $k$ to the RHS (right-hand side).
Here's a combinatorial proof. The LHS counts ways of choosing a team of $k$ people from a total of $n+1$ people, and then picking one of those $k$ to be the leader. The RHS counts the number of ways of picking one leader of the $n+1$ people, and then $k-1$ followers from the remaining $n$ people.