The total number of ways of dividing 52 cards | vedclass

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Question

(B) To divide $52$ cards into groups of sizes $17, 17, 17,$ and $1$,we first consider the number of ways to partition the set of $52$ cards into these

Vedclass · Accepted Answer

$\frac{52!}{(17!)^3}$

Vedclass · Answer

(B) To divide $52$ cards into groups of sizes $17, 17, 17,$ and $1$,we first consider the number of ways to partition the set of $52$ cards into these groups. Since $3$ groups have the same size $(17)$,we must account for the indistinguishability of these groups if we were just partitioning,but here the players are distinct. The number of ways to distribute $52$ distinct items into $4$ distinct groups of sizes $n_1, n_2, n_3, n_4$ is given by the multinomial coefficient: $\frac{52!}{n_1! n_2! n_3! n_4!}$. Here,$n_1=17, n_2=17, n_3=17, n_4=1$. Thus,the number of ways is $\frac{52!}{17! 17! 17! 1!} = \frac{52!}{(17!)^3}$.

The total number of ways of dividing $52$ cards amongst $4$ players,such that $3$ players have $17$ cards each and the fourth player has just $1$ card,is:

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