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  4. If n=5, then (nC0)2+(nC1)2+(nC2)2+..... +(nC5)2 is equal to

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Q. If $ n=5, $ then $ {{{{(}^{n}}{{C}_{0}})}^{2}}+{{{{(}^{n}}{{C}_{1}})}^{2}}+{{{{(}^{n}}{{C}_{2}})}^{2}}+..... $ $ +{{{{(}^{n}}{{C}_{5}})}^{2}} $ is equal to

KEAMKEAM 2007Binomial Theorem

Solution:

$ {{{{(}^{n}}{{C}_{0}})}^{2}}+{{{{(}^{n}}{{C}_{1}})}^{2}}+{{{{(}^{n}}{{C}_{2}})}^{2}}+.....+{{{{(}^{n}}{{C}_{5}})}^{2}} $
$={{{{(}^{5}}{{C}_{0}})}^{2}}+{{{{(}^{5}}{{C}_{1}})}^{2}}+{{{{(}^{5}}{{C}_{2}})}^{2}}+{{{{(}^{n}}{{C}_{3}})}^{3}}+{{{{(}^{5}}{{C}_{4}})}^{4}} $ $ +{{{{(}^{5}}{{C}_{5}})}^{2}} $
$=1+25+100+100+25+1 $
$=252 $