Bridge cards

How many bridge hands are possible containing 4 spades,6 diamonds, 1 club, and 2 hearts?

Result

n =  1244117160

Solution:

C4(13)=(134)=13!4!(13−4)!=13⋅12⋅11⋅104⋅3⋅2⋅1=715 C6(13)=(136)=13!6!(13−6)!=13⋅12⋅11⋅10⋅9⋅86⋅5⋅4⋅3⋅2⋅1=1716 C1(13)=(131)=13!1!(13−1)!=131=13 C2(13)=(132)=13!2!(13−2)!=13⋅122⋅1=78 s=4+6+1+2=13 n=(s4)⋅ (s6)⋅ (s1)⋅ (s2)=715⋅ 1716⋅ 13⋅ 78=1244117160=1.244117⋅109C_{{ 4}}(13) = \dbinom{ 13}{ 4} = \dfrac{ 13! }{ 4!(13-4)!} = \dfrac{ 13 \cdot 12 \cdot 11 \cdot 10 } { 4 \cdot 3 \cdot 2 \cdot 1 } = 715 \ \\ C_{{ 6}}(13) = \dbinom{ 13}{ 6} = \dfrac{ 13! }{ 6!(13-6)!} = \dfrac{ 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 } { 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 } = 1716 \ \\ C_{{ 1}}(13) = \dbinom{ 13}{ 1} = \dfrac{ 13! }{ 1!(13-1)!} = \dfrac{ 13 } { 1 } = 13 \ \\ C_{{ 2}}(13) = \dbinom{ 13}{ 2} = \dfrac{ 13! }{ 2!(13-2)!} = \dfrac{ 13 \cdot 12 } { 2 \cdot 1 } = 78 \ \\ s=4+6+1+2=13 \ \\ n={ { s } \choose 4 } \cdot \ { { s } \choose 6 } \cdot \ { { s } \choose 1 } \cdot \ { { s } \choose 2 }=715 \cdot \ 1716 \cdot \ 13 \cdot \ 78=1244117160=1.244117\cdot 10^{ 9 }

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3{2*-2}-4{2*-1}<12

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