Q:

Calculate probability of at least 1 combination happens from a series of probabilities

I understand probability well enough to get by most things but this one has me stumped. I have a pool of 25 cards total. the cards are distributed as following:
Card A: 5 copies
Card B: 6 copies
Card C: 3 copies
Card D: 1 copy
Card E: 2 copies
Card F: 8 copies

There is a draw of 5 cards. 
The odds of Card A and Card C being drawn together is 32.8%. i derived this by using :
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NeitherFig = 25 (total) - 5 (cardA) - 3 (CardC) == 17    
NoA = 25 (total) - 5 (cardA) == 20
NoC = 25 (total) - 3 (cardC) == 22    

NeitherDenom = 25 * 24 * 23 * 22 * 21 == 6375600    
NeitherNumerator = 17 (NeitherFig) * 16 * 15 * 14 * 13 == 742560    
NoANumerator = 20 (NoA) * 19 * 18 * 17 * 16 == 1860480    
NoCNumerator = 22 (NoC) * 21 * 20 * 19 * 18 == 3160080

Probability of Neither = NeitherNumerator / NeitherDenom == 742560/6375600 == 0.116    
Probability of No A = NoANumerator / NeitherDenom == 1860480/6375600 == 0.292    
Probability of No C = NoCNumerator / NeitherDenom == 3160080/6375600 == 0.496    

Probability of A and C being drawn = 1 - (Probability of No A + Probability of No C - Probability of Neither) == 1 - (.292 + .496 - .116) == .328

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I used this similar strategy to calculate the odds of other combinations to yield this table:
Card A and Card C happening = .328
Card A and Card D happening = .127
Card B and Card C happening = .367
Card B and Card E happening = .264

What i want to know is how to calculate that from this 25 card distribution, what is the probability that i draw at least 1 of these 4 combinations
i originally tried converting them back into the odds that they wouldn't happen:
Card A and Card C not happening = .672
Card A and Card D not happening = .873
Card B and Card C not happening = .633
Card B and Card E not happening = .736

and taking 1 - (product of these 4 combinations not happening)
=
1 - (.672 * .873 * .633 * .736) == .727

i designed a monte carlo simulation of this and confirmed the probability of each independent combination happening, however when i look at the number of times i got at least 1 of these 4 combinations in my first 5 cards im around 70.0% odds. I've done enough simulations to confidently say my .727 figure is wrong. Im guessing maybe it's because they aren't independent of one another. Can anyone point me in the right direction on a mathematical strategy to accurately calculate this without having to run simulations each time? 
sorry how this is worded - i've never taken a formal class on probabilities or statistics so i don't really know the terminology. 

A:

You are right that you made a calculation error by assuming events were independent (when they are not). The joint probability of X and Y, is the product of the probability of X with the probability of Y given that X also occurred, ie. $P(XY)=P(X)P(Y|X)$. Only if $X$ and $Y$ are independent can you simply multiply the probabilities, $P(XY)=P(X)P(Y)$ as you did. 
I suggest you first try a simpler case with only 2 card combinations of interest.

