Binomial Coefficient/Examples/Number of Bridge Hands

Theorem

The total number $N$ of possible different hands for a game of bridge is:

$N = \dfrac {52!} {13! \, 39!} = 635 \ 013 \ 559 \ 600$

$\dbinom {52} {13} = 2^4 \times 5^2 \times 7^2 \times 17 \times 23 \times 41 \times 43 \times 47$

The total number of cards in a standard deck is $52$.

The number of cards in a single bridge hand is $13$.

Thus $N$ is equal to the number of ways $13$ things can be chosen from $52$.

Thus:

$\ds N$

$=$

$\ds \dbinom {52} {23}$

$\ds $

$=$

$\ds \frac {52!} {13! \left({52 - 13}\right)!}$

$\ds $

$=$

$\ds \frac {52!} {13! \, 39!}$

$\ds $

$=$

$\ds 635 \ 013 \ 559 \ 600$

after calculation

$\blacksquare$