Premise: I'm trying to develop a card game to play with my friends. From a standard Western deck of 52 cards (or a double deck of 104 for more players), each player will maintain a hand of 8 cards. Players will attempt to match certain patterns (4 of a kind, 4 in a row, etc.), but always consisting of 4 cards. In order to assign point values, I want to know how common each pattern is. I've done calculations for over a dozen patterns, but I discovered by accident that I'd been using faulty logic.
Basic Example: For easy inspection, I've reduced my technique to a simple example. Suppose I draw two cards from the deck. How many possible pairs will have at least one red card? By my technique, I would start by saying there are 26 possible red cards to choose for the first card -- thus, $\binom{26}{1}$. Once that's done, there are 51 cards left, and any one will be acceptable, so the total is $\binom{26}{1}\binom{51}{1} = 1326$. Of course, this is wrong because the total number of unique pairs in the deck is $\binom{52}{2} = 1326$.
Request: For the first part, why is my logic wrong? (I have done a little research on StackExchange, and started to understand this.) For the second part of my request, is there a reasonable way to adapt the technique to get correct answers? If I would have to consider large numbers of cases individually, I might just use a computer to brute-force the solution. That would be its own fun, but lacks elegance.