How many degrees of freedom are associated with the residuals in each of the following circumstances?

(a) A completely randomized design to study three treatments with 7, 8, and 9 replicates respectively.

(b) A randomized paired comparison of two treatments A and B, where there are n_A = 15 observations with A and n_B = 15 observations with B.

(c) A full 2⁴ factorial design with three replicates. The model includes all factorial effects.

(d) An unreplicated 2⁸² factorial design with resolution V. The model includes all the main effects and two-factor interactions.

(e) A central composite design using a 2³¹ factorial design and 5 center runs. A standard second order model is fitted.