A student

A student is to answer 8 out of 10 questions on the exam.
a) find the number n of ways the student can choose 8 out of 10 questions
b) find n if the student must answer the first three questions
c) How many if he must answer at least 4 of the first 5 questions?

Correct result:

a =  45
b =  21
c =  35


C8(10)=(108)=10!8!(108)!=10921=45 a=(810)=45C_{{ 8}}(10) = \dbinom{ 10}{ 8} = \dfrac{ 10! }{ 8!(10-8)!} = \dfrac{ 10 \cdot 9 } { 2 \cdot 1 } = 45 \ \\ a={ { 8 } \choose 10 }=45
C5(7)=(75)=7!5!(75)!=7621=21 b=(83103)=21C_{{ 5}}(7) = \dbinom{ 7}{ 5} = \dfrac{ 7! }{ 5!(7-5)!} = \dfrac{ 7 \cdot 6 } { 2 \cdot 1 } = 21 \ \\ b={ { 8-3 } \choose 10-3 }=21

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