I do not understand how to do these problems.

1. 1 dice is rolled five times, find the probability that the sequence is increasing (ex. 12345, 12346, 23456, etc.)

2. 1 dice is rolled five times, find the probability that the sequence contains exactly two numbers (11234, 22314, etc.)

I can get the cardinality of the sample space. (6^5)

I'm mostly having a hard time with the cardinality of the events. There are so many possibilities for each of these, it's impossible to write them out on paper. How do I get the cardinality of all these events?

This assignment is due soon and for the most part I understand it. But I do not understand how to get these problems with the large event cardinalities.