The probability of a royal flush in a 5-card hand is 1/649740 or 0.0001539%.
The combination is a way of selecting a smaller number of sets from a larger number of sets in no particular order.
If selecting r items from a set of n items, in no particular order, we use combinations, using the formula:
nCr = n!/{r!(n - r)!}.
In the question, we are asked the probability of a royal flush in a 5 card hand.
The number of ways of selecting 5 cards from a deck can be computed using a combination, 52C5, as we have 52 cards in a deck, 5 cards are to be chosen and the order is not in concern.
52C5 = 52!/{5!(52 - 5)!} = 52!/(5!.47!) = (52*51*50*49*48)/(5*4*3*2*1) = 2598960.
The number of hands that make a royal flush is only 4. 4 from each suit.
Therefore, the probability = 4/2598960 = 1/649740 or 0.0001539%.
Therefore, the probability of a royal flush in a 5-card hand is 1/649740 or 0.0001539%.
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