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Copulas are a powerful tool to model dependence between the components of a random vector. One well-known class of copulas when working in two dimensions is the Farlie-GumbelMorgenstern (FGM) copula since their simple analytic shape enables closed-form solutions to many problems in applied probability. However, the classical definition of high-dimensional FGM copula does not enable a straightforward understanding of the effect of the copula parameters on the dependence, nor a geometric understanding of their admissible range. We circumvent this issue by studying the FGM copula from a probabilistic approach based on multivariate Bernoulli distributions. This paper studies high-dimensional exchangeable FGM copulas, a subclass of FGM copulas. We show that dependence parameters of exchangeable FGM can be expressed as convex hulls of a finite number of extreme points and establish partial orders for different exchangeable FGM copulas (including maximal and minimal dependence). We also leverage the probabilistic interpretation to develop efficient sampling and estimating procedures and provide a simulation study. Throughout, we discover geometric interpretations of the copula parameters that assist one in decoding the dependence of high-dimensional exchangeable FGM copulas.