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# Probability of Picking Groups of Cards

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Find the probability that a hand of 13 cards dealt from a well shuffled pack of 52 contains:

a) Exactly two kings and one ace
b) Exactly one ace given that it contains exactly two kings
d) At least two diamonds given that it contains exactly 3 spades and 4 hearts

https://brainmass.com/math/probability/probability-picking-groups-cards-15605

#### Solution Preview

Solution. Define C(n,k)=n!/[k!(n-k)!]. For example, C(3,2)=3!/2!=3 and C(5,2)=5!/(3!*2!)=10.

If we choose a hand of 13 cards dealt from a well shuffled pack of 52, there are C(52,13) possible cases.

(a) Exactly two kings and one ace.

There are C(4,2)*C(4,1)=6*4=24 possible case to choose exactly two kings and a ace. Then the rest of 10 cards must be chosen from those 11*4=44 cards. So there are C(44,10) possible case. So the total possible case for "exactly two kings and one ace" is ...

#### Solution Summary

Combinations are used to calculate the probability of picking groups of cards.

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