Checking the independence of compound events

Question

Given there are $n$ compound events, how many pairs of events does one have to check? As I understand it, one would have to check every pair, every triple, ... every $m$-tuple. Thus: $\sum_{k=1}^n {n \choose k}$ or $(1+1)^n -1 = \sum_{k=0}^n {n \choose k}-1 =\sum_{k=1}^n {n \choose k}=2^n -1$ times. But I have, that the correct result is $2^n-n-1$.