Question

How many ways are there to deal hands of five cards to each of six players from a deck containing 48 different cards?

Short answer

Therefore, there are\(649,352,163,073,816,339,512,038,979,194,880\)different ways

Step-by-step solution

Step 1: Assumptions and substitution

If six players are each dealt a hand of five cards from a deck of forty-eight cards, then the first player is dealt five cards, the second, third fourth, fifth and sixth player is also dealt five cards and we then have

\(\begin{array}{l}n = 48(5\; \times \;6) = 18\\k = 7\end{array}\)

\(\begin{array}{l}{n_1} = 5\\{n_2} = 5\\{n_3} = 5\\{n_4} = 5\\{n_5} = 5\\{n_6} = 5\\{n_7} = 18\end{array}\)

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 \(\begin{array}{l}\frac{{n!}}{{{n_1}!{n_2}!......{n_k}!}} = \frac{{48!}}{{5!5!5!5!5!5!18!}}\\ = 649,352,163,073,816,339,512,038,979,194,880\end{array}\)

Therefore, there are\(649,352,163,073,816,339,512,038,979,194,880\)different ways