0

Once a deck of cards is shuffled, can the order of all of the cards for that specific shuffle be determined by only knowing perhaps the order(sequence) of $4$ of the cards in the deck? How many possible shuffles would include these $4$ cards in sequence? What is the math behind it?

I would assume that the number of shuffles would be far less than 52!. I am considering options like $52*51*50*49 (6,497,400)$ possible shuffles given the 4 card sequence...

I am a very math novice.

Thank you.

User8976
  • 12,637
  • 9
  • 42
  • 107
Philip Larkins
  • 1
  • I don't think knowing the order of four cards would allow you to determine the order of all the cards in the deck. – 123 Jan 05 '15 at 17:21

1 Answers1

0

As you note, the total number of possible shufflings in a deck of cards is $52!$. This is because the first card can be any of $52$, the second can be any of the remaining $51$ cards, and so on. If you know the exact position of $4$ cards, there are still a lot of possibilities. The first unknown card can be any of the $48$ remaining cards, the second can be any one of $47$, and so on, giving a total of $48!$ possible deck arrangements that have the same four cards in the same positions. This is less than the starting number of possibilities by a factor of $6497400$, but is still far to many to state with one is correct with any reasonable degree of certainty.

If you want to know for sure what the order of all the cards is, you need to know the positions of $51$ of the cards.

KSmarts
  • 3,364