Question

How many different ways are there of dealing 52 cards to four players (Player 1, Player

2, Player 3 and Player 4) so that each player gets exactly one ace? Hint: First deal

the aces, then the rest of the cards. Simpflify.

Answer 1

Here

In a standard deck of 52 cards, we have 4 aces and 48 non-ace cards.

While dividing 52 cards among four persons, each person will get 52/4 = 13 cards.

To give each person only one ace card, he/she has to be given another 13 - 1 = 12 cards from remaining 52 -4 = 48 non-ace cards.

From these 48 cards,12 cards can be chosen by ways.

and 4 cards can be distrubuted among 4 students so that each of them will get exact 1 ace by ways

For the 1st player, the total no of ways are

Now,

For the second one between 36 nonace cards 12 cards can be distributed by ways

and between 3 aces 1 ace can be chosen by ways

So,

For the second one between 36 nonace cards 12 cards can be distributed by ways

and between 3 ace 1 ace can be chosen by ways

So,

For the 2nd one the no of ways are ways

For the 3rd one between 24 non-ace cards 12 cards can be distributed by ways

and between remaining 2 aces, 1 ace can be chosen by ways

The no of ways for the 3rd person is   ways.

Now,

For last one between 12 non-ace cards 12 cards can be distributed by ways

And last 1 ace can be given in ways

Hence ,

Total no of ways of dealing 52 cards to four players (Player 1, Player2, Player 3 and Player 4) so that each player gets exactly one ace are -