Binomial Coefficient/Examples/Number of Bridge Hands

From ProofWiki
Jump to navigation Jump to search

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

$\blacksquare$


Sources