Question 1206213: Five cards are selected from a 52-card deck for a poker hand.
(a) How many simple events are in the sample space?
simple events
(b) A royal flush is a hand that contains the A, K, Q, J, and 10, all in the same suit. How many ways are there to get a royal flush?
ways
(c) What is the probability of being dealt a royal flush? (Enter your probability as a fraction.)
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816)   (Show Source):
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Answers:
(a) 2,598,960
(b) 4
(c) 1/649740
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Explanation for part (a)
There are 52 ways to pick the first card. Then 51 choices for the next card, 50 for the next, and so on.
We keep this countdown going until five cards are selected.
52*51*50*49*48 = 311,875,200
That is the number of permutations and would be the answer if order mattered.
But order does NOT matter with a poker hand.
We must divide by 5! = 5*4*3*2*1 = 120 to get 311875200/120 = 2,598,960 which is the final answer to part (a). This value is roughly 2.6 million.
Here's another way to reach that value.
Use the nCr combination formula with n = 52 and r = 5.
n C r = (n!)/(r!(n-r)!)
52 C 5 = (52!)/(5!*(52-5)!)
52 C 5 = (52!)/(5!*47!)
52 C 5 = (52*51*50*49*48*47!)/(5!*47!)
52 C 5 = (52*51*50*49*48)/(5!)
52 C 5 = (52*51*50*49*48)/(5*4*3*2*1)
52 C 5 = (311,875,200)/120
52 C 5 = 2,598,960
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Explanation for part (b)
There is only one way to get a royal flush for any particular suit.
Since there are 4 suits, that yields 4 royal flushes possible.
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Explanation for part (c)
Divide the results of (b) over (a)
4/2598960 = 1/649740
Answer by ikleyn(52778)  (Show Source):
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