# Why number of possible cuts in a graph differ from maximum number of minimum cuts?

According to this wikipedia link, under the title "Success probability of the contraction algorithm": Number of possible cuts in a graph is $$2^{n-1}-1$$, while maximum number of minimum cuts is $${n}\choose{2}$$, where $$n$$ is number of vertices.

My question:
Why number of possible cuts differ from maximum number of minimum cuts? why isn't any cut a candidate minimum cut?

In a specific graph, not any cut is a minimum cut. What Wikipedia claims is that out of the $$2^{n-1}-1$$ potential cuts, at most $$\binom{n}{2}$$ can be minimum cuts in any given graph. Some graphs contain even fewer minimum cuts. For example, if you connect two cycles by a single edge then there is a unique minimum cut.
The upper bound on the number of minimum cuts follows by analyzing Karger's algorithm (linked to in the question): you can show that the probability that the algorithm produces any specific minimum cut is at least $$1/\binom{n}{2}$$, and so there can be at most $$\binom{n}{2}$$ of them.