# 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?

Result

p =  1.29 %

#### Solution:

$a = ({ { 1 } \choose 4 } )^{ 3 } = 64 \ \\ C_{{ 1}}(4) = \dbinom{ 4}{ 1} = \dfrac{ 4! }{ 1!(4-1)!} = \dfrac{ 4 } { 1 } = 4 \ \\ \ \\ C_{{ 3}}(32) = \dbinom{ 32}{ 3} = \dfrac{ 32! }{ 3!(32-3)!} = \dfrac{ 32 \cdot 31 \cdot 30 } { 3 \cdot 2 \cdot 1 } = 4960 \ \\ \ \\ b = { { 3 } \choose 32 } = 4960 \ \\ p = 100 \cdot \ a /b = 100 \cdot \ 64 /4960 = \dfrac{ 40 }{ 31 } \doteq 1.2903 = 1.29 \%$

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