If `.^(n+2)C_8:^(n-2)P_4:: 57 : 16 ,` find `n`

Explanation

`Rightarrow frac{\ ^{n+2} C_{8}}{n-2}=frac{57}{16}`

`Rightarrow frac{(n+2) !}{8 !(n-6) !} times frac{(n-6) !}{(n-2) !}=frac{57}{16}`

`Rightarrow frac{(n+2)(n+1) n(n-1)(n-2) !}{8 !} times frac{1}{(n-2) !}=frac{57}{16}`

`Rightarrow(n+2)(n+1) n(n-1)=frac{57}{16} times 8 !=frac{19 times 3}{16} times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1`

`Rightarrow(n+2)(n+1) n(n-1)=143640`

`Rightarrow(n-1) n(n+1)(n+2)=19 times 3 times 7 times 6 times 5 times 4 times 3`

`Rightarrow(n-1) n(n+1)(n+2)=19 times(3 times 7) times(6 times 3) times(4 times 5)`

`Rightarrow(n-1) n(n+1)(n+2)=18 times 19 times 20 times 21`

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