Thirteen cards and 635,013,559,600 possibilitie ][

Actually a remake of my previous article. In that article I was trying to calculate the total number of different possible games that can be seen in bridge.

Now, I’ll explain how I arrive to the same number 635,013,559,600 – which is the total number of possible different hands you can receive.

The Basics

If you pick a hand of 1 card out of a 52 card deck, there’s 52 possible different cards.

If you try to picks a second card out of the remaining 51 decks, there’s an extra 51 possibilities.

If you combine both these possibilities, you’ll do it as so: 52 x 51 = 2652 possible different 2-card hands that can be drawn out of a 52 card deck.

If you draft the whole 52 cards, you’ll go 52x52x50… all the way down to 1. There is a notation for that called, factorial and is noted as such : 52!.

A Bit More

There is no easy notation for multiplying from 52! down to, lets say, 25!. However, since it’s multiplications, we can do a division to ease our eyes from rowfulls of numbers. 52!/24!will give us the exact same thing as if we were going 52x51x50x…x26x25 (we divided out the whole 24x23x22x21..x1 part). Neat.

So we could say that a formula to calculate the number of possible hands would go as such:

p = d! / (d – h)!

p is the number of possibilities
d is the number of cards in our deck
h is the number of cards in our hand

But that’s not exactly precise.

More Precision

The number of possibilities we calculated for when you draft a 2 card hand from a 52 card decks does not take into account the number of similar hands you can draft.

It counts the hand {ace of clubs’, king of spades} as different from {king of spades, ace of clubs »}, which is not true for our purpose.

The easy way out of this is to calculate the number of ways a single hand can be composed.

And that’s quite easy. A 2-hand card has exactly… 2×1 possibile combinations (2!)

So we could say that out of the 2651 possible combinations, only 1326 (also known as 2652 / 2) are unique hands.

The Formula

To integrate this into our formula:

p = d! / (d – h)! / h!

p is the number of possible unique hands
d is the number of cards in our deck
h is the number of cards in our hand

Have fun and play with it – I made a nice JavaScript that does it for you 😛

Cards in deck:
Cards in hand:
Possible unique hands: