Question 1111164: In how many ways can a 3 card hand be dealt from a standard deck of 52 card if all are to be face cards?
We know there are 12 face cards in a deck of 52 therefore
(12!3!) using combination = 220?
Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website! Yes.
Just pointing out the notation is usually 12C3 (usually the 12 and 3 would be subscripted, though many variations of notation exist for combinations).
nCr = n! / ((n-r)!*r!) <—— combinations, n-choose-r: used when order does not matter.
12C3 = 12! / (9!*3!) = 12*11*10 / (3*2) = 220.
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The long answer/explanation:
There are 12 possible face cards for the first card, 11 for the 2nd card, and 10 for the 3rd card. This gives 12*11*10 = 1320 possible _draws_ of 3 face cards. However, we're not done. Looking closely, there is an over counting here. For example, being dealt { Qh, Ks, Jc } is counted as distinct from { Ks, Jc, Qh } even though these are the same hand. Thus, we must figure out how much over counting there is and divide that out. Three cards can be arranged in 3! ways, so we must divide our earlier answer by 3!: 1320/3! = 1320/6 = 220. Exactly the same result we obtained above. In fact, the r! in the denominator of the nCr formula is exactly the division we just did.
I hope this helps!
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