It’s easy - calculate 54!, then divide by 53 and 54.

52! is the number of possible arrangements of a deck of cards.

To see why, let’s say you pick cards at random from one deck and add them to another, initially empty, deck. First card can be any of 52, second card can be any of 51 remaining, and so on until there is only one card left. So total number of possible arrangements is $52 \cdot 51 \cdot ... \cdot 3 \cdot 2 \cdot 1 = 52!$

To calculate 52! by hand, start with Stirling’s approximation $n!=\sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$ for $n=54$, then divide by 53 and 54:

$$52! = \frac{54!}{53\cdot54} = \frac{\sqrt{2 \pi 54}\left(\frac{54}{e}\right)^{54}}{53\cdot54}$$

Now you should see why we started with 54 rather than 52. The convenient coincidence that $54 = 20 \cdot 2.7 \approx 20 e$ can be used to simplify calculations.

$$52! \approx \frac{\sqrt{110\pi}\cdot20^{54}}{2500} \approx \frac{\sqrt{400}\cdot2^{54}\cdot10^{54}}{2500}$$

Finally, multiply numerator and denominator by 4 and use the fact that $2^{10}\approx 10^3$ to get

$$52! \approx \frac{2\cdot 10\cdot 10^{15}\cdot2^{6}\cdot10^{54}}{10^4} = 2^{7} \cdot 10^{66} \approx 10^{68}$$

The exact value is 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 or about $8 \cdot 10^{67}$.

Source: http://www.solipsys.co.uk/new/Calculating52FactorialByHand.html