Your notations are too complicated and make it harder to see a solution. Your problems mix problems known as:

The Birthday Paradox
The Classical Occupancy Problem
Ion Saliu’s Paradox of N Trials

You can find solutions and also SOFTWARE here:
http://saliu.com/monty-paradox.html
http://saliu.com/birthday.html
http://saliu.com/theory-of-probability.html