How many combinations of cards are possible with one shuffle?

Factual Questions
1

Just one shuffle. For the purpose of the question:

Assume that you cut the deck exactly in half.
You do only one pass.

I suspect it’s somewhat less than 52! because the cards in each half will always be in exactly the same order wherever they appear in the new deck.

2

This is C(52,26) – the number of ways of selecting 26 out of 52 slots.
C(52,26) = 52!/(26!*26!)
Google will respond to the latter expression with 4.9591853e+14, i.e. almost 500 trillion.

This thinking leads you directly to the formula. 52! must be divided by 26! (because 26 of the cards have a fixed internal ordering) divided by another 26! (because the other 26 do also).

3

Thanks. I thought it was something simple.

4

Note that this is allowing as a “shuffle” operations such as “all of one half of the deck hits the table, and then all of the other half hits on top of it”. If you put more constraints on what counts as a shuffle, then the number will go down, but we’d have to know exactly what constraints you’re using.

5

I assume this is the classic one-shot interleave or side shuffle. The order of each pile is predetermined, each pile is exactly 26 cards.

the only questions is - first card - A1 or B1? next card - Next available A or B? etc.
This is analogous to - how many different ways can you flip a coin and get 26 heads in 52 or less flips? (If less than 52 flips, fill in the rest as Tails).