Thread: Choosing 4 cards of different suits and different values from a deck of 52 cards.

Hello All,

Please help me solve this problem:
How many ways are there to choose 4 cards of different suits and different values from a deck of 52 cards?

Thank you,
Ashwin Kumar D P, Bangalore, India

Hello, Ashwin!

I had to "draw a picture" to understand this one . . .

How many ways are there to choose 4 cards of different suits
and different values from a deck of 52 cards?

There are, of course, 52 cards in the deck.

. .


The first card can be any of the 52 cards.
. . Suppose it is
Then these choices remain for the second card:

. .

There are 36 choices for the second card.


Suppose the second card is
Then these choices remain for the third card:

. .

There are 22 choices for the third card.


Suppose the third card is
Then these choices remain for the fourth card:

. .

There are 10 choices for the fourth card.


Disgarding the order in which the four cards are drawn,

. . there are: . such 4-card hands.

I understand where the 52, 36, 22, and 10 came from, but why divide by 4!? could anyone describe it in a way that makes sense?