Number of ways of dividing 52 cards amongst four players so that three players have 17 cards each an

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6.Permutation and Combination

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The number of ways of dividing $52$ cards amongst four players so that three players have $17$ cards each and the fourth player just one card, is

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

$52\;!$

$\frac{{52\;!}}{{17\;!}}$

None of these

(IIT-1979)

Solution

(a) For the first set number of ways $^{52}{C_{17}}$.

Now out of $35$ cards left $17$ cards can be put for second in $^{35}{C_{17}}$ ways similarly for $3$ rd in $^{18}{C_{17}}$.

One card for the last set can be put in only one way.

Therefore the required number of ways for the proper distribution

$ = \frac{{52\,!}}{{35\,!\,17\,!}} \times \frac{{35\,!}}{{18\,!\,17\,!}} \times \frac{{18\,!}}{{17\,!\,1\,!}} \times 1\,! = \frac{{52\,!}}{{{{(17\,!)}^3}}}$.

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