I've got a dataset with 3 factor variables with only one interaction. `Y` is the response and `A`,`B`,`C` are the independent variables.  Variable `B` in particular has a factor that occurs with no other factors, so the design matrix has a 0 in it, leading to an error when trying to run a mixed model (fixed factors only).

I'm trying to figure out how to reduce number of variables so the design matrix doesn't have a 0 in it.

In the case where all variables and all possible interactions are included, this is fairly straightforward, as I can create one new variable with all possible factor values for `A`,`B`,`C`.  I've tested this in R and get equivalent values with: `Y ~ A * B * C` (which expands to `Y ~ A + B + C + A:B + A:C + B:C + A:B:C`) provides the same output as `Y ~ D` when `A`,`B`,`C` are put together in a new variable `D`.  Basically I'm reducing it to a one way design.

Idea taken from here – 
https://stats.stackexchange.com/questions/89780/can-you-convert-three-way-anova-to-one-way-anova

However, what I need to do is only explore two interactions, not all of them.  The formula I'm trying to use is `Y ~ A + B + C + A:B + A:C`.

Can anybody recommend how to reduce this to fewer factor variables so I don't end up with a 0 in the design matrix?  I've been trying a lot of combinations but am mentally not able to wrap my head around this.