How many edges must a graph with N vertices have in order to guarantee that it is connected?

here on stack overflow they said (n-1)*(n-2)/2 +1

https://cs.stackexchange.com/questions/7373/how-many-edges-must-a-graph-with-n-vertices-have-in-order-to-guarantee-that-it-i

but i have doubt in it suppose we consider graph on 10 vertices.

here we need just 21 edges for n=10

but according to given formula it is 9*8/2 + 1 = 37.

so what is correct?

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