one pair

Number of combinations to pick 4 cards randomly from a pack of playingcards
= C(52,4)
= 52!/4!48!
= 270725

To get 4 cards with exactly "one pair", select any one card number (fromthe 13 card numbers) and pick two cards of this card number (each card numberhas 4 suits) , then select two other card numbers (from the rest 12 cardnumbers) and pick one cards of each of the two card numbers.

Number of combinations to pick 4 cards getting exactly one pair
= C(13,1) x C(4,2) x C(12, 2) x C(4, 1)²
= 13 x (4!/2!2!) x (12!/10!2!) x 4²
= 82368

The required probability
= 82368/270725
= 6336/20825