In how many ways can 52 cards be distributed (i) Equally among 4 players and , (ii) Equally into 4 groups ?

Explanation

(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))`