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  4. Find nC21 if nC10=nC11.

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Q. Find $ ^{n}{{C}_{21}} $ if $ ^{n}{{C}_{10}}{{=}^{n}}{{C}_{11}}. $

J & K CETJ & K CET 2014Permutations and Combinations

Solution:

Given, $ ^{n}{{C}_{10}}{{=}^{n}}{{C}_{11}} $
$ \Rightarrow $ $ \frac{n!}{10!(n-10)!}=\frac{n!}{11!(n-11)!} $
$ \Rightarrow $ $ 11\times 10!(n-11)!=0!(n-10)(n-11)! $
$ \Rightarrow $ $ 11=n-10 $
$ \Rightarrow $ $ n=11+10=21 $ Now, $ ^{n}{{C}_{21}}{{=}^{21}}{{C}_{21}}=1 $