So my question has two parts and I have managed to confused myself into lack of comprehension.
Suppose you are handed a shuffled, stacked deck of playing cards i.e. instead of 52 cards there are 104 as the I included an extra copy of each card for the heart suit (two 2 of hearts, two 3 of hearts, etc) and four copies of each spades (four 2 of spades, four 3 of spades, etc).
(P(hearts) = 1/4, P(spades)=1/2, P(clubs)=P(diamond)=1/8)
Pick a number, n. What is the likelihood that in the top n cards there are h hearts, s spades, and c clubs? where h + s + c + d = n (where d is the number of diamonds)
A bit harder: Now pick two numbers m and n. First remove the first m cards from the randomized deck. Now what is the likelihood that in the top n cards there are h hearts, y spades, and c clubs, h + s + c + d = n (where d is the number of diamonds).
So for the first part (without removing m cards at random) is clear to me that the top n cards of the stacked deck of cards is 104Cn an the number of ways the specific combination of (h, s, c, d) is the multinomial coefficient (n!/(h!*s!*c!*d!)). Where I am confusing myself is, is this just the (n C h,s,c,d / 104Cn * 100) as 104Cn inherently handles the uneven distributions in the suits of cards? or do I need to weight the multinomial coefficient somehow?