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The likelihood that an undercooled liquid vitrifies or crystallizes depends on the cooling rate R. The critical cooling rate Rc, below which the liquid crystallizes upon cooling, characterizes the glass-forming ability (GFA) of the system. While pure metals are typically poor glass formers with Rc > 1012 K/s, specific multi-component alloys can form bulk metallic glasses (BMGs) even at cooling rates below R ~ 1 K/s. Conventional wisdom asserts that metal alloys with three or more components are better glass formers (with smaller Rc) than binary alloys. However, there is currently no theoretical framework that provides quantitative predictions for Rc for multi-component alloys. In this manuscript, we perform simulations of ternary hard-sphere systems, which have been shown to be accurate models for the glass-forming ability of BMGs, to understand the roles of geometric frustration and demixing in determining Rc. Specifically, we compress ternary hard sphere mixtures into jammed packings and measure the critical compression rate, below which the system crystallizes, as a function of the diameter ratios sB/sA and sC/sA and number fractions xA, xB, and xC. We find two distinct regimes for the GFA in parameter space for ternary hard spheres. When the diameter ratios are close to 1, such that the largest (A) and smallest (C) species are well-mixed, the GFA of ternary systems is no better than that of the optimal binary glass former. However, when sC/sA ≲ 0.8 is below the demixing threshold for binary systems, adding a third component B with sC < sB < sA increases the GFA of the system by preventing demixing of A and C. Analysis of the available data from experimental studies indicates that most ternary BMGs are below the binary demixing threshold with sC/sA < 0.8. [ABSTRACT FROM AUTHOR]