Question 1201199: How many ways to create a five card poker hand with 2 aces and 2 kings?
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3817)   (Show Source):
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Answer: 1584
Explanation:
Let's say the first card we want is neither an ace nor a king.
There are 4 aces and 4 kings.
That's 4+4 = 8 cards total.
There are 52-8 = 44 cards that are neither an ace nor a king.
That's the number of choices we have for the first slot.
Put another way: For any given suit, there are 13-2 = 11 cards that are neither an ace nor a king.
Four suits total yields 4*11 = 44 cards that are neither an ace nor a king.
Now to count the number of ways to pick the two kings.
There are n = 4 kings to pick from and r = 2 slots to fill.
Order does not matter with poker hands.
This means we go for the nCr combination formula.
n C r = (n!)/(r!(n-r)!)
4 C 2 = (4!)/(2!*(4-2)!)
4 C 2 = (4!)/(2!*2!)
4 C 2 = (4*3*2!)/(2!*2!)
4 C 2 = (4*3)/(2!)
4 C 2 = (4*3)/(2*1)
4 C 2 = (12)/(2)
4 C 2 = 6
There are 6 ways to select the two kings.
This value can be found in Pascal's Triangle.
Put another way: We have 4*3 = 12 permutations if order mattered.
But order doesn't matter so we divide by 2 to get 12/2 = 6 ways to pick the two kings.
Since this value is fairly small, we can list all the ways to pick the two kings.- KC,KD
- KC,KH
- KC,KS
- KD,KH
- KD,KS
- KH,KS
where,
KC = king of clubs
KD = king of diamonds
KH = king of hearts
KS = king of spades
Order doesn't matter so a group like KC,KD is the same as KD,KC.
Using practically identical logic, you should find that there are 6 ways to pick the two aces.
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Recap:
- 44 ways to pick a card that isn't an ace, and isn't a king either
- 6 ways to pick two kings
- 6 ways to pick two aces
That will give 44*6*6 = 1584 different five card poker hands consisting of 2 aces and 2 kings.
Order does not matter with poker hands.
Answer by greenestamps(13200)  (Show Source):
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