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How many ways can you shuffle a deck of cards, so that the Ace of Hearts is not directly on top of the King of Hearts?

 Feb 19, 2021
 #1
avatar+31264 
+1

There are 52! ways to arrange the 52 cards of the deck

   there are only 51 ways to have the ace of hearts directly on top of the king of hearts in the shuffled deck positions

     but that leaves the other 50 cards to arrange in any order or  50!

 

 

52!  - 51 * 50!= 52!-51! =

        79107056417656496291436393839934463764226245720866291712000000000000 = 7.9107056417656496e67

 ( I think !)

..

 Feb 19, 2021
edited by ElectricPavlov  Feb 19, 2021
 #2
avatar+117978 
+1

Let  us  look at the number  of  ways in which  the ace  of  hearts   could  be  directly on the king of  hearts

 

These  two  cards  could occupy   51 positions.....and  for  each of  these, the other  cards  could  be  arranged in 50! ways

 

The total  number of  arrangements=  52!

 

So....the  number of ways  that  the  ace of hearts is not directly on the  king of hearts   =

 

52!  -  51 * 50!    =   52!  - 51!    =     EP's  answer    !!!!

 

 

cool cool cool

 Feb 19, 2021

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