Question 1203217
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Answer: <font color=red size=4>0.144%</font> (approximate)



Explanation


Use the nCr combination formula to determine how many five card hands are possible. Order does not matter.


n = 52 cards
r = 5 selections
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 = 311875200/120
52 C 5 = 2598960


There are 2,598,960 ways to select five cards in any order.


Many calculators have a button labeled "nCr" to make such calculations quick and painless.


Here is one online calculator
<a href="https://www.gigacalculator.com/calculators/combination-calculator.php">https://www.gigacalculator.com/calculators/combination-calculator.php</a>


The next step is to determine how many full house hands are possible.


Focus on the example trio 9♠, 9♥, 9♦
(ie 9 of spades, 9 of hearts, 9 of diamonds)
There's 1 card left out (9 of clubs). We do this to avoid getting 4 of a kind.
There are 4 ways to pick the unlucky card that gets left out.


Also there are 13 choices for the rank of this trio.
That gives 13*4 = 52 ways to form the trio.


For the duo (2♣, 2♥) we have 2 cards left out.
There are 4C2 = 6 ways to determine the suits for this duo. And there are 12 remaining ranks to pick from.
We have 6*12 = 72 ways to select the duo.


We found there are...<ul><li>52 ways to form the trio</li><li>72 ways to form the duo (after the trio is formed)</li></ul>It yields 52*72 = 3744 different full house hands.
Again, order does not matter.


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In summary there are:<ul><li>3744 ways to get a full house</li><li>2,598,960 ways to select any five cards</li></ul>Order does not matter.


Divide the values
3744/(2,598,960) = 0.0014405762305
This value is approximate.


Multiply by 100 to convert to a percent.
This moves the decimal point 2 spots to the right.


0.0014405762305 = 0.14405762305%


That rounds to <font color=red size=4>0.144%</font> when rounding to 3 decimal places.


Further Reading:
<a href="https://en.wikipedia.org/wiki/Poker_probability">https://en.wikipedia.org/wiki/Poker_probability</a>
and 
<a href="https://math.stackexchange.com/questions/808314/probability-of-getting-a-full-house">https://math.stackexchange.com/questions/808314/probability-of-getting-a-full-house</a>
and 
<a href="https://socratic.org/questions/592b200311ef6b2839d30feb">https://socratic.org/questions/592b200311ef6b2839d30feb</a>
and 
<a href="https://math.stackexchange.com/questions/4153143/how-many-ways-are-there-to-get-a-full-house">https://math.stackexchange.com/questions/4153143/how-many-ways-are-there-to-get-a-full-house</a>
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