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  4. 20C4+2⋅20C3+ 20C2-22C18 is equal to :

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Q. $ {}^{20}C_{4}+2\cdot{}^{20}C_{3}+ {}^{20}C_{2}-{}^{22}C_{18} $ is equal to :

Jharkhand CECEJharkhand CECE 2005

Solution:

If $ n $ and $ r $ positive integer and $ 0\le r < n $ ,
then $ {}^{n}C_{r}+{}^{n}C_{r-1}={}^{n+1}C_{r} $
$ \therefore {}^{20}C_{4}+2.{}^{20}C_{3}+{}^{20}C_{2}-{}^{22}C_{18} $
$ =({}^{20}C_{4}+{}^{30}C_{3})+({}^{20}C_{3}+{}^{20}C_{2})-{}^{22}C_{18} $
$ ={}^{21}C_{4}+{}^{21}C_{3}-{}^{22}C_{18} $
$ ={}^{22}C_{4}-{}^{22}C_{18} $
$ ={}^{22}C_{18}-{}^{22}C_{18} $
$ =0 $