Two groups

The group of 10 girls should be divided into two groups with at least 4 girls in each group. How many ways can this be done?

Result

n =  462

Solution:

C4(10)=(104)=10!4!(10−4)!=10⋅9⋅8⋅74⋅3⋅2⋅1=210  C5(10)=(105)=10!5!(10−5)!=10⋅9⋅8⋅7⋅65⋅4⋅3⋅2⋅1=252  n=(104)+(105)=462C_{{ 4}}(10)=\dbinom{ 10}{ 4}=\dfrac{ 10! }{ 4!(10-4)!}=\dfrac{ 10 \cdot 9 \cdot 8 \cdot 7 } { 4 \cdot 3 \cdot 2 \cdot 1 }=210 \ \\ \ \\ C_{{ 5}}(10)=\dbinom{ 10}{ 5}=\dfrac{ 10! }{ 5!(10-5)!}=\dfrac{ 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 } { 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 }=252 \ \\ \ \\ n={ { 10 } \choose 4 } + { { 10 } \choose 5 }=462

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