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!(104)!=109874321=210  C5(10)=(105)=10!5!(105)!=10987654321=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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