(i) In how many ways can a pack of 52 cards be divided equally among four players? (ii) (ii) In how many ways can you divide these cards in four sets, three of them having 17 cards each and the four the one just one card?

Options

A. `(52!)/((17!)^(3)3!)`
B. `(52!)/((17!)^(3))`
C. `(52!)/(51!(17!)^(3))`
D. `(52!)/(51!)`

Explanation

First we divide 52 cards into two groups of 1 card and 51 cards. This can be done in `(52!)/(51!1!)` ways. Now, every group of 51 cards can be divided into 3 groups of 17 card each in `(51!)/((17!)^(3)3!)`
`:.` Number of ways `=(52!)/(51!1!)xx(51!)/((17!)^(3)3!)=(52!)/((17!)^(3)3!)`
ALITER For the first set we have `""^(52)C_(17)` choices, for the second set `""^(35)C_(17)` choices, for the third set `""^(18)C_(17)` choices and for the last set we have `""^(1)C_(1)` choices. But the first three sets can be interchanged in 3! ways.
`:.` Total number of ways of distribution
`=""^(52)C_(17)xx""^(35)C_(17)xx""^(18)C_(17)xx"^(1)C_(1)xx(1)/(3!)=(52!)/((17!)^(3)3!)`