# Binomial Coefficient/Examples/Number of Bridge Hands

## Theorem

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

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

### Prime Factors

The prime decomposition of the number of bridge hands is given as:

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

## Proof

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:

 $\displaystyle N$ $=$ $\displaystyle \dbinom {52} {23}$ Cardinality of Set of Subsets $\displaystyle$ $=$ $\displaystyle \frac {52!} {13! \left({52 - 13}\right)!}$ Definition of Binomial Coefficient $\displaystyle$ $=$ $\displaystyle \frac {52!} {13! \, 39!}$ $\displaystyle$ $=$ $\displaystyle 635 \ 013 \ 559 \ 600$ after calculation

$\blacksquare$