From a Standard 52 card deck of playing cards. If 10 hands are dealt consisting of 2 cards each what is the probability of 3 of the hands being pairs? 4? 5? ....
Is there a formula for this?
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Hint: $$\frac{52!}{2!^{10}}$$ is the number of ways to select 10 sets 2 cards each. Can you handle from here?
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We need to draw only 20 cards.... – true blue anil Aug 20 '15 at 07:04
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'10 sets 2 cards each'= 20 cards – Alex Aug 20 '15 at 07:35
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52! means dealing all the cards, but only 20 are needed – true blue anil Aug 20 '15 at 08:35
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Use multinomial identity – Alex Aug 20 '15 at 11:47
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Hint:
How many ranks are there ?
Choose 3 ranks, and pairs from them.
How many cards remain of a rank other than those of the pairs chosen ?
Choose 7 hands with 2 "singles" each from them
$$\text{Favorable ways =}{13\choose3}{4\choose2}^3{10\choose 7}{4\choose1}^7$$
$$\text{Total ways =}\frac{{52\choose20}\cdot20!}{10!\cdot(2!)^{10}}$$
Divide to get the probability and work out similarly for 4,5 pairs
true blue anil
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