I'm trying to calculate flush frequencies given different deck compositions. From a full deck I understand the number of times a player would be dealt exactly 5 clubs to be $$\binom{13}{5}\binom{39}{2}$$ where 39 is the number of cards of other suits.
I'm struggling understanding why a similar construct is not used when calculating the number of times a player will be dealt 5 clubs given a single 7 card, 7 rank hand, i.e. 2, 3, 4, 5, 6, 7 and 8. Using this construct I would calculate as $$\binom{7}{5}\binom{21}{2}$$ where there are 21 non-club cards, any 2 of which can be used. I know this solution is not correct.
I believe the proper way to calculate this would be $$\binom{7}{5}\binom{3}{1}^2=189$$ I ultimately need to be able to calculate for different deck compositions (for example assume the 3 diamonds and 4 spades were not available) so much prefer to use a solution based on the total number of non-club cards from which to draw the remaining 2 cards, if at all possible.
If this calculation cannot be based on the total number of non-club cards remaining, what would the correct formula be in the last example, where neither the 3 of diamonds and 4 or spades are available?