### Instruction: 3 cards are chosen at random from a standard 52-card deck.  What is the probability that they form a pair?  (A 3-card hand is a 'pair' if two of the cards match in rank but the third card is different.  For example, 668 is a pair, but 999 is not.).### Response: I need to find the number of ways to form a pair out of 3 cards, and divide that by the total number of ways to choose 3 cards from 52.
To choose 3 cards from 52, I can use the combination formula: 52 choose 3, which is 52! / (3! * 49!) = 22100.
To form a pair, I need to choose a rank for the pair, then choose two cards of that rank, then choose a different rank for the third card, then choose one card of that rank.
There are 13 possible ranks for the pair, from Ace to King.
For each rank, there are 4 choose 2, which is 4! / (2! * 2!) = 6, ways to choose two cards of that rank.
For the third card, there are 12 remaining ranks to choose from, and 4 cards of each rank.
So, the number of ways to form a pair is 13 * 6 * 12 * 4, which is 3744.
Therefore, the probability of forming a pair is 3744 / 22100, which simplifies to 144 / 850, or about 0.169.