Determine $n$ if
$^{2 n} C_{3}:^{n} C_{3}=11: 1$
Determine $n$ if
$^{2 n} C_{3}:^{n} C_{3}=11: 1$
$\frac{^{2 n} C_{3}}{^{n} C_{3}}=\frac{11}{1}$
$\Rightarrow \frac{(2 n) !}{3 !(2 n-3) !} \times \frac{3 !(n-3) !}{n !}=11$
$\Rightarrow \frac{(2 n)(2 n-1)(2 n-2)(2 n-3) !}{(2 n-3) !} \times \frac{(n-3) !}{n(n-1)(n-2)(n-3) !}$
$\Rightarrow \frac{2(2 n-1)(2 n-2)}{(n-1)(n-2)}=11$
$\Rightarrow \frac{4(2 n-1)(n-1)}{(n-1)(n-2)}=11$
$\Rightarrow \frac{4(2 n-1)}{n-2}=11$
$\Rightarrow 4(2 n-1)=11(n-2)$
$\Rightarrow 8 n-4=11 n-22$
$\Rightarrow 11 n-8 n=-4+22$
$\Rightarrow 3 n=18$
$\Rightarrow n=6$
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