Statistics/Combination problem I need help with


#1

Hey all, was wondering if anyone was a genius at this subject and could help me out with this one part.

(13C3)(3C1)(4C3)(4C1)(4C1)

(13C1)(4C3)(12C1)(4C1)(11C1)(4C1) divide by 2.

These represent the number of possible 3 of a kind in 5-card poker. They BOTH equal 54,912 when calculated which is correct.

I have to explain WHY they both represent what they do. I’m stuck on the second one… I don’t know why you have to divide by 2.


#2

here’s what the second way represents: first select a face for the 3-of-a-kind. then choose 3 of the 4 suits. now choose 2 faces for the two singles. from each face, choose a suit. this is actually (13C1)(4C3)(12C2)(4C1)(4C1)=(13C1)(4C3)(12C1)(4C1)(11C1)(4C1)/2. the reason you need to divide by 2 is because without it, you’re double counting the number of ways you can select the two single cards. for instance, suppose from (12C1)(4C1) you get a King of hearts, and from (11C1)(4C1) you get a Queen of spades. well this hand is the same as if you had gotten the Queen from (12C1)(4C1) and the King from (11C1)(4C1). so you’re counting the same hand twice. dividing by 2 takes care of that.


#3

OHHHHH. Lol. Thanks man. Now I’m gonna sit here and ponder for a bit and try to really understand it. Kind of got confused because since this was a combination, I figured it would have accounted for all possible ways already.