Question 1206213
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Answers:
(a) <font color=red size=4>2,598,960</font>
(b) <font color=red size=4>4</font>
(c) <font color=red size=4>1/649740</font>


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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 = <font color=red>2,598,960</font> 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 = <font color=red>2,598,960</font>



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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 = <font color=red>1/649740</font>
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