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.