What is the probability that the hand is a three of a kind? A three of a kind has 3 cards of the same value. The other two cards do not have the same value as each other and do not have the same value as the three with the same rank. For example, {4♠, 4♦, 4♣, J♠, 8♥} is a three of a kind.
There are 13 ways to select the rank for the three of a kind. Once the rank has been selected for the three of a kind, there are C(4, 3) ways to select the three cards from the four cards with that rank. There are 12 ranks remaining for the other two cards. There are C(12, 2) ways to pick the ranks for the two other cards. Once the ranks have been chosen for those cards, there are four ways to pick the suit for the first card and four ways to pick the suit for the second card. The total number of 5-card hands that are a three of a kind is 13 C(4, 3) C(12, 2)⋅4⋅4.The probability that a random 5-card hand is a three of a kind is
13C(4, 3) C(12, 2)⋅4⋅4 / C(52, 5) = 88 / 4165