In how many ways can 52 cards be divided equally among four players(1,2,3,4) in order.
I interpreted this question as
Let there be 52 different cards(1.e each numbered with unique number from 1 to 52,like c1,c2,c3,c4....,c52).
In how many ways can cards be divided equally among four players(1,2,3,4) in order(i.e Player 1 gets card(c1,c5,c9....),Player 2 gets(c2,c6,c10...),Player 3 gets(c3,c7,c11....),Player 4 gets(c4,c8,c12....))
In my textbook it's answer is given as
For the first player we have C(52,13) choices,for the second player we have C(39,13) choices, for the third player we have C(26,13) choices and for the last player we have C(13,13) choices . Hence the total number of ways
= C(52,13)C(39,13) C(26,13)* C(13,13)
But,
According to me each player gets unique set of cards.Therefore,all will have C(13,13) ways. Hence total number of ways
=C(13,13)*C(13,13)*C(13,13)*C(13,13) =1
Where am I wrong?
You are not wrong, you are just answering a different question than what the textbook is answering. You are counting the number of ways to achieve 13 cards to each player where the value of each card received does not matter. The textbook is counting the number of possible distributions of those cards (where the order of the cards within a player's hand does not matter, but the specific cards received does). You are treating player 1 getting all of the spades as the same as if player 1 got a random assortment of cards (since in both cases, player 1 still received 13 cards). The textbook is treating those as separate outcomes.