Cards

From a set of 32 cards we randomly pull out three cards. What is the probability that it will be seven king and ace?

Correct result:

p =  1.29 %

Solution:

C1(4)=(41)=4!1!(41)!=41=4 C3(32)=(323)=32!3!(323)!=323130321=4960 a=((14))3=64 b=(332)=4960 p=100 a/b=100 64/4960=4031=1.29%C_{{ 1}}(4) = \dbinom{ 4}{ 1} = \dfrac{ 4! }{ 1!(4-1)!} = \dfrac{ 4 } { 1 } = 4 \ \\ C_{{ 3}}(32) = \dbinom{ 32}{ 3} = \dfrac{ 32! }{ 3!(32-3)!} = \dfrac{ 32 \cdot 31 \cdot 30 } { 3 \cdot 2 \cdot 1 } = 4960 \ \\ a=({ { 1 } \choose 4 } )^{ 3 }=64 \ \\ b={ { 3 } \choose 32 }=4960 \ \\ p=100 \cdot \ a /b=100 \cdot \ 64 /4960=\dfrac{ 40 }{ 31 }=1.29 \%



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