Suppose that in a company there are three groups of employees:
- $80$ associate consultants
- $50$ consultants
- $20$ senior consultants
The president of the company is calling out names of each of the $150$ employees one by one and giving them a ticket to the Universal Studios. The names of the employees are called out at random without any specific preference to any one group of employees and each employee has a different name. A group is considered "completed" if all the members of the group receive the ticket. Find the probability that the group of associate consultants gets "completed" before consultants and senior consultants.
I thought of finding the favorable cases by considering say after calling out the K-th name, the associate consultants group gets completed. So there should be atleast one person each of consultant and senior consultant in remaining 150-K names that should be called out. Then I summed over all possible values of K, but neither did this result in a closed form solution nor do I think that it is correct because cases are being repeated.