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In how many ways can a pack of 52 cards ...

In how many ways can a pack of 52 cards be
(i) Distributed equally among four players in order?
(ii) Divided into four groups of 13 cards each?

Text Solution

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(i) Each player is to get 13 cards. First player gets 13 cards out of 52 in `""^(52)C_(13)` ways. Second player gets 13 cards out of 39 cards in `""^(39)C_(13)` ways. Third player gets 13 cards out of 39-13=26 cards in `""^(26)C_(13)` ways and the fourth player gets 13 cards out of 26-13=13 cards in `""^(13)C_(13)` ways. Hence by multipliction principle, total number of ways in which cards can be distributed
`=""^(52)C_(13).^(39)C_(13).^(26)C_(13).^(13)C_(13)`
`=(52!)/(4!(13!)^(4))`

(ii) In this case order of group is not important. Hence the number of ways in which 52 different cards can be distributed equally into four groups
`(52!)/(4!(13!)^(4))`
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