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In the field of design of experiments, factorial designs are widely used to study the effects of multiple factors simultaneously. A full factorial experiment in a $2^k$ design investigates all possible combinations of $k$ factors, each at two levels, usually denoted as low $(-1)$ and high $(+1)$. For instance, a $2^2$ factorial design involves two factors, say $A$ and $B$, each at two levels, leading to $2^2 = 4$ treatment combinations. These four combinations are $(--), (-+), (+-), (++))$. Such an arrangement allows the experimenter to estimate the main effects of both factors and their interaction effect. However, in practice, as the number of factors increases, the number ________ ________ _____ ___ _______ ________ ______ ________ ___ ___ __________.
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