TAOCP 1.2.6 Exercise 3

Section 1.2.6: Binomial Coefficients

Exercise 3. [00] How many bridge hands (13 cards out of a 52-card deck) are possible?

Verified: no
Solve time: 1m25s

A bridge hand consists of 13 cards chosen from a standard deck of 52 cards. By definition, the number of ways to choose $k$ objects from $n$ distinct objects is given by the binomial coefficient $\binom{n}{k}$, as in equation (2) of Section 1.2.6. Therefore, the number of bridge hands is

$$ \binom{52}{13} = \frac{52\cdot 51 \cdot 50 \cdots 40}{13\cdot 12 \cdot 11 \cdots 1}. $$

This product has $13$ factors in both the numerator and the denominator, corresponding to the $13$ successive selections of cards for the hand. Evaluating or leaving in factorial form, we have

$$ \binom{52}{13} = \frac{52!}{13!,39!}. $$

Hence the total number of distinct bridge hands is

$$ \boxed{\frac{52!}{13!,39!}}. $$

This completes the solution.

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