Question

How many ways are there to deal hands of seven cards to each of five players from a standard deck of 52 cards?

Short answer

Therefore, there are\(69,731,208,959,871,249,835,089,602,560,000\)different ways

Step-by-step solution

Step 1: Assumptions and substitution

If five players are each dealt a hand of seven cards from a deck of 52 cards, then the first player is dealt seven cards, the second player is dealt seven cards and third players seven, the fourth and fifth players also seven cards, so we have

\(\begin{array}{l}n = 52 - (5 \times 7) = 17\\k = 6\end{array}\)

\(\begin{array}{l}{n_1} = 7\\{n_2} = 7\\{n_3} = 7\\{n_4} = 7\\{n_5} = 7\\{n_6} = 17\end{array}\)

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 \(\begin{array}{l}\frac{{n!}}{{{n_1}!{n_2}!......{n_k}!}} = \frac{{52!}}{{7!7!7!7!7!17!}}\\ = 69,731,208,959,871,249,835,089,602,560,000\end{array}\)

Therefore, there are\(69,731,208,959,871,249,835,089,602,560,000\)different ways