Question

How many ways are there to order the cards in a standard 52 card deck, such that the ace of hearts and the ace of diamonds are adjacent, but the ace of spades and the ace of clubs are not adjacent?

Answer 1

here there are 2! ways ace of hearts and the ace of diamonds are adjacent and consider as 1 unit, remaining 48 cards with them can be considered 49 cards total except that of  ace of spades and the ace of clubs. number of ways to arrange them =2!*49!

Now we have 49 cards and we need to find 2 places from 50 places in and out for ace of spades and the ace of clubs

therefore total number of ways =2!*49!*50*49=98*50!

(Another way to solve this:

Number of ways  ace of hearts and the ace of diamonds are adjacent =2!*51! (consider  ace of hearts and the ace of diamonds as one unit)

number of ways  ace of hearts and the ace of diamonds are adjacent as well ace of spades and the ace of clubs are adjacent ) =2!*2!*50! (consider those as one units)

therefore Number of ways  ace of hearts and the ace of diamonds are adjacent but  ace of spades and the ace of clubs are not adjacent =2!*51!-2!*2!*50!

=2*51*50!-2*2*50!

=50!*(102-4) =98*50! )