Minimum number of edges in a graph of order n

What is the minimum number of edges in a $k$ -chromatic connected graph of order $n$ ?

I have read somewhere that this number is equal to $\left(\begin{array}{c} k \\2 \end{array} \right) +n -k$

I was thinking about using $K_k$ , the complete graph with $k$ vertices. We know that the number of edges in $K_k$ is equal to $\left(\begin{array}{c} k \\2 \end{array} \right)$ .

Also, I wanted to use the fact that if the chromatic number of a graph $M$ , $\chi (M)$ is equal to $k$ , and if a vertex $v$ has degree strictly less than $k$ , then the chromatic number of the graph ( $M$ together with $v$ and its edges) is also equal to $k$ .

I am not sure how start, I need help organizing my ideas.