How many 5 card hands out of a 52 card deck are there that have at least one red card. I am aware of the correct answer, which is:
$$
\text{All possible hands} - \text{All hands with no red cards} = \text{All hands with at least one red card}
$$
However, there is an incorrect argument that I cannot figure out why is wrong. Here is the argument: We first select $26 \choose 1$ red card and then we select $51 \choose 4$ of the remaining cards in the deck. This produces an incorrect result and I do not know why. Intuitively it makes sense to me. Could someone help me out?

The problem is over-counting.
Consider the case where you end up with A-hearts, 2-hearts, and 3 black cards.  You count this twice, once where the Ace is the (first) red card chosen, and once where the 2 is the first red card chosen.
The actual analysis of what it would take to correct and use such a direct approach is very complicated, since you could have $k$ red cards, where $k \in \{1,2,3,4,5\}.$
Naturally, the correct enumeration is 
$\binom{52}{5} - \binom{26}{5}.$
