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Now you have 50 cards left in the deck and you must pick one of 48 "good" ones b/c if you pick another of our pair cards you will get 3 of a kind.
Now you have 49 cards left in the deck and you must not match your pair cards or the card you just selected (two pair). So that leaves 44 "good" ones and 5 bad ones.
Finally, you have 48 cards left in the deck and you must pick a non pair card. There are still 2 left from your original pair that you don't want, and there are now 3 left from your last card that you picked and 3 left from the one before that. So you have 48-8=40 good ones left.
Therefore 48*44*40/6 = 14080 ways to get the other three cards without pairing or making three of a kind (don't forget to divide by 3! to eliminate duplicate arrangements).
So in closing you have 78*14080 = 1,098,240 ways to make exactly one pair. Divide this by 52c5 = 2,598,960 to get Prob = 42.2%.