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Question

# In a game of poker, five players are each dealt 5 cards from a 52-card deck. How many ways are there to deal the cards?

Solution

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Given:

$n_1=n_2=n_3=n_4=n_5=5$

$n=52$

Definition multinomial coefficients: The number of ways in which ways $n$ objects can be grouped into $r$ classes with $n_i$ the size of the $i$th class are

$\left(\begin{matrix}n \\ n_1\: n_2 \: .... \: n_r\end{matrix}\right)=\dfrac{n!}{n_1!n_2!...n_r!}$

with $k!=k\times (k-1)\times (k-2)\times ....\times 1$.

We know that five players are each dealt 5 of the 52 cards, the number of cards remaining are then:

$n_6=n-n_1-n_2-n_3-n_4-n_5=52-5-5-5-5-5=27$

Using the definition of multinomial coefficients, we then obtain:

$\left(\begin{matrix}52 \\ 5\: 5 \: 5\: 5\: 5 \: 27\end{matrix}\right)=\dfrac{52!}{5!5!5!5!5!27!}=297,686,658,367,751,290,178,415,114,240$

Thus there are 297,686,658,367,751,290,178,415,114,240 possible ways to dealt the cards.

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