Since we have removed one card from the deck, there are 51 choices for the second card, but the choices available depend on the first card.

How many combinations of 2 cards are there?

2!52

1326

How many possible values would we have if we used base-51 for the second digit?

52 * 51

2652

This is twice the number of values that we need. It's a perfectly fine coding scheme, but it leaves gaps.

Is there an alternative?

52 * (51r2) NB. 51r2 is the fraction 51/2, or 25.5

1326

J is perfectly happy to work with fractional bases.

51r2 52 #. 0, 0 NB. lowest 2-digit number in this base

0

51r2 52 #. (51r2-1),51 NB. highest 2-digit number in this base

1325

We can convert any pair of cards back and forth to our fractional base:

51r2 52 #: 1234 51r2 52 #: 678

23 38 13 2

51r2 52 #. 23 38 51r2 52 #. 13 2

1234 678