Different ways to make the first two down cards in Texas Holdem:

C52₂ = 1,326

Ways of getting the next five up cards:

C50₅ = 2,118,760

Ways of getting the first three up cards (the flop):

C50₃ = 19,600

Ways of getting the fourth up card (the turn):

C47₁ = 47

Ways of getting the last up card (the river):

C46₁ = 46

Ways of getting the last two up cards (after the flop):

C47₂ = 1,081

Check calculation:

C50₃ * C47₂ / C5₂ = C50₅ = 19,600 * 1,081 / 10 = 2,118,760 OK.
The C5₂ term comes about because there are that many ways the same 5 cards can be arranged to come out in 3 card then 2 card sequence.

Ways of getting the entire seven cards:
C52₇ = 133,784,560 = C52₂ * C50₅ / C7₂ = 133,784,560 (see above for C7₂ term)