A poker hand consists of five cards drawn from a deck of 52

A poker hand consists of five cards drawn from a deck of 52 cards. Each card has one of 13 denominations (2, 3, 4, ..., 10, Jack, Queen, King, Ace) and one of four suits (Spades, Hearts, Diamonds, Clubs). Determine the probability of drawing a poker hand consisting of one pair (two cards of one denomination and three cards of distinct denominations, where each of the three cards has a different denomination than the denomination of the pair). The probability of drawing a poker hand consisting of one pair is. (Type an integer or decimal rounded to four decimal places as needed.)

Solution

Similar to arguments for previous hands there are 13 ranks to choose from for the pair and 4C2 possible pairs per rank, plus (48 * 44 * 40)/6 ways to choose the other three cards (again to remove permutations and keep only combinations I must divide by 3!, the number of permutations of the three cards). This leaves 13 * 4C2 * (48 * 44 * 40)/6 = 1,098,240 possible one pair hands