Question

In bridge, the 52 cards of a standard deck are dealt to four players. How many different ways are there to deal bridge hands to four players?

Short answer

Therefore, there are\(53,644,737,765,488,729,839,237,440,000\)different ways

Step-by-step solution

Step 1: Assumptions and substitution

If 52 cards are dealt to four players, then each of the four player will get 13 cards.\(\begin{array}{l}n = 52\\k = 4\end{array}\)

\(\begin{array}{l}{n_1} = 13\\{n_2} = 13\\{n_3} = 13\\{n_4} = 13\end{array}\)

Step 2: Further simplification

Distributing n distinguishable objects into k distinguishable boxes such that \({n_i}\) objects are placed in box i (\(i = 1,2,3,4,5\)) can be done in \(\begin{array}{l}\frac{{n!}}{{{n_1}!{n_2}!......{n_k}!}} = \frac{{52!}}{{13!13!13!13!}}\\ = 53,644,737,765,488,729,839,237,440,000\end{array}\)

Therefore, there are\(53,644,737,765,488,729,839,237,440,000\)different ways