A factorial design has four factors, $A$, $B$, $C$, and $D$, with , , , and levels, respectively. The response $y$ of interest is observed once for every combination of the levels of the 4 factors. Let $Y_{ijkm}$ be a random variable representing possible values of the observation $y_{ijkm}$ for level $i$ of $A$, level $j$ of $B$, level $k$ of $C$, and level $m$ of $D$. Assume the following linear model: \begin{align*} Y_{ijkm} & = \mu + \alpha_i + \beta_j + \gamma_k + \delta_m \\ & + (\alpha\beta)_{ij} + (\alpha\gamma)_{ik} + (\alpha\delta)_{im} \\ & + (\beta\gamma)_{jk} + (\beta\delta)_{jm} + (\delta\gamma)_{km} \\ & + \epsilon_{ijkm}, \end{align*} where the $\epsilon_{ijkm}$ are assumed to be independent ${\mbox{N}}(0, \sigma^2)$ random variables. Note that this model includes main effects and 2-factor interaction effects but no higher-order interaction effects.

Part a) How many degrees of freedom (df) are there for the $A$ main effect? Give your answer as an integer.

Part b) How many df are there for the $B \times C$ 2-factor interaction effect? Give your answer as an integer.

Part c) How many df are there for the residuals? Give your answer as an integer.