Can I use polling techniques in card games? In a 50 card deck where 4 copies of a card are allowed at maximum,there are around 2 million possible starting hands.
My question is: If I take a random sample of the 300-350 and analyze it. Can I draw conclusions about the 2 million starting hands?
For example: If in my sample the 20% of starting hands have 2 copies of Card A and 1 of Card B. Can I infer that 20% of starting hands from the 2 million possibilities will have 2 copies of Card A and 1 of Card B?

Let's take this to the extreme: you look at a random subset of 100 hands from the 2M possible combinations and find out that it had a specific winning hand. Could you conclude that 1% of the combinations have it? No, it's the opposite: if you found this specific hand in your subset, it means that it does not appear in the remaining combinations because it's unique. You are dealing with a finite population and need to keep it in mind.
If you are sampling without replacement (take a random subset of all the hands) each sample you took depletes the pool of available hands in the pool. To avoid such problems, you should sample the hands with replacement. Such sampling would fit better for answering the question of "what you could expect if you picked the hand at random".
For finite samples, you would also have specialized statistics, e.g. finite population corrections, and models, like hypergeometric distribution. You would need them if you sampled without replacement.
Finally, in your example, you could use a computer to list and count all the possible combinations in just a few lines of code.
