
A QUICKER SOLUTION:
OBSERVATIONS:
 The number of cards in each suit is different.
 The number of hearts and diamonds is 5, so there can only be the following possible pairs:
 1 heart + 4 diamonds
 2 hearts + 3 diamonds
 3 hearts + 2 diamonds
 4 hearts + 1 diamond
 So the maximum number of diamonds or hearts is 4
 The number of diamonds and spades is 6, so there can only be the following possible pairs:
 1 diamond + 5 spades
 2 diamonds + 4 spades
 (cannot be 3 + 3)
 4 diamonds + 2 spades
 (cannot have more than 5 diamonds)
 The numbers of diamonds or spades cannot be 3
 The number of clubs is the remainder of the 13 cards.
Let us "diagram" the solution:

