No, CBC is a completely different animal. In contrast to traditional full-profile conjoint, we aren't asking for a rating (on a scale, such as a 100-point scale) of individual profiles. Rather, we are showing respondents sets of multiple profiles, and asking for a single choice (1/0) of one of the profiles. Thus, CBC asks respondents to consider more information than a single card at a time, yet the answer itself provides less statistical information whereby to estimate utilities. We know which concept the respondent selected, but we don't know by how much it was preferred over the others, or the relative ranking of the non-selected concepts.
There has been much debate and multiple articles presented at our conferences on the "right" number of choice tasks to ask in CBC. So much does depend on the number of parameters to estimate, which is given (at least for main effects) by the formula you cite above (assuming you are using a None category, which accounts for the extra intercept parameter). But, if you decide that you want to include interaction effects between two attributes in the model, then you have even more parameters to estimate.
I wish there were an easy formula to point to to tell you exactly how many questions to include in any CBC questionnaire. But, typically CBC/HB estimation is used, which makes it possible to estimate a full set of utilities for each respondent even under the most sparse of conditions (due to its ability to leverage data from the population). So, even with just a few choice tasks per respondent, CBC/HB will indeed produce a full set of utilities for the respondent (not saying they'll be very accurate... just that it will produce a full set of utilities for each respondent). With very few tasks, each respondent's utilities will resemble the sample's means quite a bit, because much "smoothing" to the global mean needed to occur to compensate for the uncertainty at the individual level. The more tasks one includes, the more accurate the utilities will be for each individual... probably up to a point, and then the accuracy will decrease as respondent fatigue/error sets in.
The "typical" approach for the "typical" CBC study is to ask each respondent from 8 to 20 choice tasks.
Here are two listings of articles on the subject:
And, see the article by Tang/Grenville in https://www.sawtoothsoftware.com/download/techpap/2010Proceedings.pdf