The value of ^100C_5051+^100C_5152+ .+^100C_100101 is:

Question

The value of $\frac{{}^{100}{C}_{50}}{51}+\frac{{}^{100}{C}_{51}}{52}+\ldots .+\frac{{}^{100}{C}_{100}}{101}$ is:

Accepted Answer

$\begin{gathered} (1+x{)}^{100}={}^{100}{C}_{0}+{}^{100}{C}_{1x}+{}^{100}{C}_{2{x}^{2}}+\ldots \ldots ..+{}^{100}{C}_{100{x}^{100}} \ \sum _{r=50}^{100}\frac{{}^{100}{C}_{r}}{r+1}=\frac{1}{101}\sum _{r=50}^{100}{}^{101}{C}_{r+1} \ \Rightarrow \frac{1}{101}\sum _{r=51}^{100}{}^{101}{C}_{r} \ \Rightarrow \frac{{2}^{100}}{101} \end{gathered}$

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