SOLUTION: How many poker hands consist of two cars of one ranked two cards of another different Rank and one card of a third-ranked (such a poker hand is called two pairs)

Question 1137515: How many poker hands consist of two cars of one ranked two cards of another different Rank and one card of a third-ranked (such a poker hand is called two pairs)
Answer by math_helper(2461) (Show Source): You can put this solution on YOUR website!
Using notation nCr = n!/((n-r)!r!)

Form of hand: aabbc (where a,b,c represent different RANKS of any suit)

(#ways to draw two a's)*(#ways to draw two b's)*(#ways to draw c)

Notes: 13C2 ways to draw a,b from the 13 ranks
4C2 ways to arrange rank a (4 suits, 2 chosen)
4C2 ways to arrange rank b (4 suits, 2 chosen)
11C1 ways to draw c from the remaining 11 ranks
4C1 ways to draw c of a given suit

Putting it together:
13C2*4C2*4C2*11C1*4C1 = 78*6*6*11*4 = hands
----
But, you may ask, why not pick the unmatched card (c) from the 13 ranks first? Let's try it that way:

13C1 ways to pick rank c
4C1 ways to draw c of a given suit
12C2 ways to pick ranks a,b <<< note how there are 12 ranks to pick from
4C2 ways to arrange rank a suits
4C2 ways to arrange rank b suits

= 13*4*66*6*6 = 123552, as before

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