A deck of cards contains 3 kids of face cards, Jack, Queen, and King 
of 4 different suits, which is 12 face cards.  Therefore it contains
52-12 or 40 non-face cards

The hand may contain any of 12C4 or 495 different groups of 4 face cards.
The hand's 5th non-face card could be any of the 40 non-face cards.  So
the number of possible hands that contain exactly 4 face cards and 1 non-
face card is 495×40 = 19800

The total number of possible 5-card poker hands is 52C5 = 2598960

So the probability is 19800/2598960 

Reduce by dividing top and bottom by 120

Answer = 165/21658, or about 0.0076

Edwin