Doubt in monty hall problem discussed on Mini Exam #10

While explaining the monty hall problem, greg explained all three cases. But is it possible to have another case where the magician picks up c3 instead of c2 in b x x scenario? If that’s true, then the total number of cases will be 4 and the probability will be 1/2.
For better understanding, I have attached an image of my solution.

Didn’t attend the session but this article explains the solution to that problem:

https://www.statisticshowto.com/probability-and-statistics/monty-hall-problem/