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Question

If $$(n + 2)C_{8} : (n - 2)P_{4} = 57 : 16$$, find $$n$$.


Solution

Given that,

$$\dfrac { \left( n+2 \right) { C }_{ 8 } }{ \left( n-2 \right) { P }_{ 4 } } =\dfrac { 57 }{ 16 } $$ 

$$ \Longrightarrow \dfrac { \dfrac { (n+2)! }{ (n-6)!8! }  }{ \dfrac { (n-2)! }{ (n-6)! }  } =\dfrac { 57 }{ 16 } $$ 

$$ \Longrightarrow \dfrac { (n+2)(n+1)(n)(n-1) }{ 8! } =\dfrac { 57 }{ 16 } $$ 

$$ \Longrightarrow (n+2)(n+1)(n)(n-1)=143640=21.20.19.18$$ 

$$ \Longrightarrow n=19$$

Mathematics

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