Can someone check with me if my answer is right for this question? I got $n(n+1)/2$ . The question is posted below. $\newcommand{\Barbell}{\operatorname{Barbell}}$
For $n$ an integer with $n ≥ 3$, let $\Barbell(n)$ be the graph that is made by attaching with a single edge two complete graphs with $n$ vertices each. A complete graph with $n$ vertices is a graph in which every vertex is adjacent to every other vertex besides itself. The following question is about $\Barbell(n)$. For an integer $n ≥ 3$, how many edges are in the graph $\Barbell(n)$?
To clarify, given $n\geq 3$ you are given two graphs $G_{1}$ and $G_{2}$, each of which is a copy of the complete graph on $n$ vertices, $K_{n}$. The graph Barbell$(n)$ is then formed by choosing a vertex $v_{1}\in G_{1}$ and a vertex $v_{2}\in G_{2}$ and then adding an edge between $v_{1}$ and $v_{2}$ to connect $G_{1}$ and $G_{2}$ together.
Hint:
How many edges are in $K_{n}$?
$K_{2}$ has $1$ edge.
$K_{3}$ has $3$ edges.
$K_{4}$ has $6$ edges.
In general there are as many edges in $K_{n}$ as there are ways of choosing a subset of size $2$ out of the set $\{1,2,\ldots,n\}$.
If the number of edges in $K_{n}$ is $E(n)$ then the answer to your problem is $2E(n)+1$.