SOLUTION: How many ways to create a five card poker hand with 2 aces and 2 kings?

Algebra.Com
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): You can put this solution on YOUR website!

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.
  1. KC,KD
  2. KC,KH
  3. KC,KS
  4. KD,KH
  5. KD,KS
  6. 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.

-------------

Recap:
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): You can put this solution on YOUR website!


We need to choose 2 of the 4 aces, 2 of the 4 kings, and 1 of the other 44 cards:



ANSWER: 1584
RELATED QUESTIONS

What is the probability of drawing 3 kings and 2 aces in a 5 card hand of poker? (answered by VFBundy)
What is the probability of having two aces, two kings, and a queen in a five-card poker... (answered by stanbon)
How many 5-card poker hands consisting of 3 aces and 2 kings are possible with an... (answered by stanbon)
How many 5-card poker hands consisting of 2 aces and 3 kings are possible with an... (answered by jorel1380)
How many 5-card poker hands consisting of 3 aces and 2 kings are possible with an... (answered by Boreal)
Calculate the probability of getting a five card poker hand consisting of 3 kings and 2... (answered by Fombitz)
Compute the probability that a five-card poker hand is dealt to you that contains 2 Aces. (answered by math_tutor2020)
A poker hand consists of five cards randomly dealt from a standard deck of 52 cards. The... (answered by greenestamps,math_tutor2020)
Compute the number of a 5-card poker hands containing 2 any Aces, 2 any Kings, and 1 any... (answered by Edwin McCravy)