Quote:
Originally Posted by Theoretical
(24/52) x (15/51) x (8/50) x 10
Besides kapw's correction shown in bold, you made the mistake of counting the same possibilities multiple times. You allowed the 4th and 5th card to be anything, and some of those anythings are cards lower than 6. When you multiply it all by 10, the possibilities of 4 or 5 cards lower than 6 get counted more than once, which is why your answer is too high. Overlap is a bitch.
To avoid that, you can either split it into cases like Whosnext did, or you can use inclusion-exclusion (affectionately referred to as PIE in this forum). I'm a big fan of PIE, but I don't think it saves time for this problem, due to the fact that different possibilities are over-counted by varying magnitudes (at least, at a glance I think they are). PIE is handy when the over-counting is uniform; if it were uniform here, the problem would be reduced to 3 steps (as opposed to the 9 shown by Whosnext). But if it's not uniform, PIE can be as tedious as the direct method, sometimes more so.