Finding RHS of combinatorial proof: $n^2=(n-2)^2+2(n-2)+2n$

Question

The identity: $n^2=(n-2)^2+2(n-2)+2n$

The question that I’m answering is “how many lists of length 2 on n elements are there.”

LHS is fairly obvious in terms of answering the question, but I’m not sure how to answer the RHS, combinatorially.

Answer 1

HINT: Draw an $(n-2)×(n-2)$ square. Then add two rows of length $n-2$ each, to the top of the $(n-2)×(n-2)$ square, to get a $(n-2)×n$ rectangle. Then add $2$ columns of length $n$ each, to the right of the $(n-2)×n$ rectangle to get an $n×n$ square....

Answer 2

Let the elements be $\{1,\dots,n\}$, and consider three disjoint cases:

• The list contains neither $n-1$ nor $n$.
• The list contains exactly one of $\{n-1,n\}$ and in the first position.
• The list contains either $n-1$ or $n$ in the second position.