The given experiment involves three factors $($armature length, spring load, and bobbin depth$)$ with three levels $($low$,$ medium, and high$)$ for each factor, with a single observation on flow rate for each combination of levels. We want to determine how many observations are in the resulting data set, that is$,$ the total number of factor$-$level combinations. For example, "armature length$-$low, spring load$-$low, bobbin depth$-$medium" is one such factor$-$level combination, so the flow rate for this combination is one observation.

The following table lists some of the possible combinations.

Armature Length Spring Load Bobbin Depth

Low Low Low

Low Low Medium

Low Low High

Low Medium Low

Consider the combinations that have "armature length$-$low" and "spring load$-$low." There are three such combinations, corresponding to each level option for bobbin depth. These options for bobbin depth are repeated for each level option of spring load, while keeping the factor$-$level option "armature length$-$low." Thus, there are

$33=$

combinations of spring load and bobbin depth levels that have "armature length$-$low."