Combinatorics - In how many ways can three aces and two kings be drawn

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1. In how many ways can three aces and two kings be drawn?

2. There are 24 ways three aces can be drawn and there are 12 ways two kings can be drawn.
3. I tried 24x12=288 ways to get three aces and two kings.. people are telling me that it is wrong, but I am not understanding why
 

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HallsofIvy
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1. In how many ways can three aces and two kings be drawn?

2. There are 24 ways three aces can be drawn and there are 12 ways two kings can be drawn.
3. I tried 24x12=288 ways to get three aces and two kings.. people are telling me that it is wrong, but I am not understanding why
First look at specifically "AAAKK" in that order. There are 4 aces so 4 ways to draw that first ace. After that, there are 3 aces left so 3 ways to get the second ace. After drawing the second ace, there are 2 aces left so 2 ways to draw that third ace. There are 4 kings so 4 ways to draw the first king. Then there are 3 kings left so 3 ways to draw that second king.

So far that says there are 4(3)(2)(4)(3)= 24(12)= 288 ways to get "AAAKK" in that specific order. But the same kind of analysis would show that there are 288 ways to get, say, "AKAKA" or three aces and two kings in any specific order. You need to multiply 288 by the number of ways to order 3 "A"s and 2 "K"s. Do you know how to find that?
 
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First look at specifically "AAAKK" in that order. There are 4 aces so 4 ways to draw that first ace. After that, there are 3 aces left so 3 ways to get the second ace. After drawing the second ace, there are 2 aces left so 2 ways to draw that third ace. There are 4 kings so 4 ways to draw the first king. Then there are 3 kings left so 3 ways to draw that second king.

So far that says there are 4(3)(2)(4)(3)= 24(12)= 288 ways to get "AAAKK" in that specific order. But the same kind of analysis would show that there are 288 ways to get, say, "AKAKA" or three aces and two kings in any specific order. You need to multiply 288 by the number of ways to order 3 "A"s and 2 "K"s. Do you know how to find that?
Wow I would like to say thank you, because you actually explained it in a manner I can understand.
On your question, "the number of ways to order 3 "A"s and 2 "K"s. Do you know how to find that?" I would use, n!/(n-r)!= 5!/(5-3)!=5!/2!=60 ways to order 3 "A" cards out of 5 cards.
Then n!/(n-r)!=5!/(5-2)!=5!/3!=20 ways to order 2 "k" cards out of 5 cards.
Then should I add 20 +60= 80 total ways to order 3"A"s and 2"k"s?
 

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