I find hypergeometric series nice for computing, but poor for explaining. For teaching purposes, I much prefer expanding these out using binomial coefficients. Some books, and Wolfram Alpha call this "choose."
The key here is to realize that order does not matter. The total number of possible hands is choose(104, 10) = 26 trillion.
Now, to count how ...
How does one calculate the probability of hands in a card game, when the deck involved has 8 suits?
This depends on the query you seek, and the specific restrictions it might have. Answering your example:
I imagine that calculating the probability of my hand containing two aces (of any suits) is the simplest starting point for this game's rules.