Quote from EricP:
Just for completeness, let me do a quick calculation of what the odds would be to get four aces after the flop (i.e. after five cards).
Note that getting this would involve getting one useless card, that wasn't an ace. This makes calculation of the odds a little more complex. For the sake of simplicity, the notation for an ace will be "A" and the notation for any other card (2 - K) will be "x".
Also, note that you can get four aces after the flop in five ways:
AA AAx (we'll call this "I")
AA AxA (II)
AA xAA (III)
Ax AAA (IV)
xA AAA (V)
To correctly figure the overall odds, we must calculate the individual odds of each of these scenarios (I through V) and sum them up to get the total overall odds.
Odds of getting scenario I:
(4/52)*(3/51)*(2/50)*(1/49)*(48/48) = 1/270,725
Odds of getting scenario II:
(4/52)*(3/51)*(2/50)*(48/49)*(1/48) = 1/270,725
Odds of getting scenario III:
(4/52)*(3/51)*(48/50)*(2/49)*(1/48) = 1/270,725
Odds of getting scenario IV:
(4/52)*(48/51)*(3/50)*(2/49)*(1/48) = 1/270,725
Odds of getting scenario V:
(48/52)*(4/51)*(3/50)*(2/49)*(1/48) = 1/270,725
=> Overall odds of getting four aces after the flop would be the sum of all of these which would be 5/270,725, or one chance out of 54,145.
From this, you can see that it is much 'easier' to get four aces (1:54,145) than it is to get a royal flush (1:6.5 million).
-Eric