# Test

The teacher prepared a test with ten questions. The student has the option to choose one correct answer from the four (A, B, C, D). The student did not get a written exam at all. What is the probability that:

Correct result:

a =  0.058
b =  0
c =  0.056

#### Solution:

$C_{{ 5}}(10) = \dbinom{ 10}{ 5} = \dfrac{ 10! }{ 5!(10-5)!} = \dfrac{ 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 } { 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 } = 252 \ \\ p=1/4=\dfrac{ 1 }{ 4 }=0.25 \ \\ q=1-p=1-0.25=\dfrac{ 3 }{ 4 }=0.75 \ \\ n=10 \ \\ n_{1}=n/2=10/2=5 \ \\ a={ { n } \choose n_{1} } \cdot \ p^{ n_{1} } \cdot \ q^{ n-n_{1} }=252 \cdot \ 0.25^{ 5 } \cdot \ 0.75^{ 10-5 }=0.058$
$C_{{ 10}}(10) = \dbinom{ 10}{ 10} = \dfrac{ 10! }{ 10!(10-10)!} = \dfrac{ 1 } { 1 } = 1 \ \\ b={ { n } \choose n } \cdot \ p^{ n } \cdot \ q^{ n-n }=1 \cdot \ 0.25^{ 10 } \cdot \ 0.75^{ 10-10 }=0$
$C_{{ 0}}(10) = \dbinom{ 10}{ 0} = \dfrac{ 10! }{ 0!(10-0)!} = \dfrac{ 1 } { 1 } = 1 \ \\ c={ { n } \choose 0 } \cdot \ p^{ 0 } \cdot \ q^{ n-0 }=1 \cdot \ 0.25^{ 0 } \cdot \ 0.75^{ 10-0 }=0.056$

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