A standard deck of cards (with no jokers) consists of 4 suits (clubs, spades, he

Question

A standard deck of cards (with no jokers) consists of 4 suits (clubs, spades, hearts, and diamonds), each of which contain 13 kinds of cards (2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace). In five-card draw poker, each player is dealt a hand of 5 different cards from this deck. IF the hand contains 3 cards of the same kind and the remaining 2 cards are from 2 different kinds, this hand is referred to as "three of a kind". What is the probability that the hand you are dealt will be a "three of a kind"?

Explanation / Answer

Number of ways of drawing 5 cards from 52 cards = 52C5
Now for "three of a kind", we have different cases
3 cards are one kind (say club), 1 card is of other kind (say spade), 1 card is of another kind (say diamond)
since we have 3 kinds of cards at a time
Number of ways in which we can 3 kinds of cards = 4C3
3 cards of same kind can be selected in = 13C3 ways
1 card of same kind can be selected in = 13C1 ways
another 1 card of same kind can be selected in = 13C1 ways
Total number of ways for "three of a kind" to satisfy is = 4C3 x 13C3 x 13C1 x 13C1
Probability for "three of a kind" is = [4C3 x 13C3 x 13C1 x 13C1] / 52C5

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