The game of euchre uses only 9s, 10s, jacks, queens, kinds, and aces from a standard deck of cards. How many five-card hands have:

(a) at least two red cards?

My attempt:

12C2 x 22C3 = 101640

12C2 represent the two red card

22C3 represent all the card leftover after removing the two red cards. And since the restriction is met it wouldn't matter if I mix the red and black card together... right?

I understand that this question can be solved by:

All combination(24C5) - no red(12C5) - 1 red(12C1x12C4), and the answer should be 35772

At the same time, I don't understand what is wrong with my original logic, please help