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  4. Determine n, if 2nC2:nC2: = 9:2

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Q. Determine $n$, if $ ^{2n}{{C}_{2}}{{:}^{n}}{{C}_{2}}:\,\,\,\,=\,\,9:2 $

J & K CETJ & K CET 2014Permutations and Combinations

Solution:

Given, $ ^{20}{{C}_{2}}{{:}^{n}}{{C}_{2}}=9:2 $
$ \Rightarrow $ $ \frac{^{20}{{C}_{2}}}{^{n}{{C}_{2}}}=\frac{9}{2} $
$ \Rightarrow $ $ \frac{(2n)!}{2!(2n-2)!}\times \frac{2!(n-2)!}{n!}=\frac{9}{2} $
$ \Rightarrow $ $ \frac{(2n)(2n-1)(2n-2)!}{2(2n-2)!}\times \frac{2(n-2)!}{n(n-1)(n-2)!}=\frac{9}{2} $
$ \Rightarrow $ $ \frac{2(2n-1)}{n-1}=\frac{9}{2} $
$ \Rightarrow $ $ 4(2n-1)=9(n-1) $
$ \Rightarrow $ $ 8n-4=9n-9 $
$ \Rightarrow $ $ 9n-8n=-4+9 $
$ \Rightarrow $ $ n=5 $