Classroom

Of the 26 pupils in the classroom, 12 boys and 14 girls, four representatives are picked to the odds of being:
a) all the girls
b) three girls and one boy
c) there will be at least two boys

Correct result:

p₁ =  0.067
p₂ =  0.292
p₃ =  0.641

Solution:

$p_{1}=\dfrac{ 14 \cdot \ 13 \cdot \ 12 \cdot \ 11 }{ 26 \cdot \ 25 \cdot \ 24 \cdot \ 23 }=\dfrac{ 77 }{ 1150 }=0.067$

$C_{{ 3}}(14)=\dbinom{ 14}{ 3}=\dfrac{ 14! }{ 3!(14-3)!}=\dfrac{ 14 \cdot 13 \cdot 12 } { 3 \cdot 2 \cdot 1 }=364 \ \\ \ \\ C_{{ 1}}(12)=\dbinom{ 12}{ 1}=\dfrac{ 12! }{ 1!(12-1)!}=\dfrac{ 12 } { 1 }=12 \ \\ \ \\ C_{{ 4}}(26)=\dbinom{ 26}{ 4}=\dfrac{ 26! }{ 4!(26-4)!}=\dfrac{ 26 \cdot 25 \cdot 24 \cdot 23 } { 4 \cdot 3 \cdot 2 \cdot 1 }=14950 \ \\ \ \\ p_{2}={ { 14 } \choose 3 } \cdot \ { { 12 } \choose 1 } / { { 26 } \choose 4 }=364 \cdot \ 12 / 14950=\dfrac{ 168 }{ 575 }=0.292$

$p_3=1-p_1-p_2=1-0.067-0.2922={\displaystyle \frac{641}{1000}}=0.641$

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