Evaluate the following: ^5_r=1 ^5C_ r - Vidyadip

Question

Evaluate the following: $\sum\limits^5_\text{r=1}\ ^5\text{C}_{\text{r}}$

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Answer

We have, $= { ^5}\text{C}_{\text{1}}+{ ^5}\text{C}_{\text{2}}+{ ^5}\text{C}_{\text{3}}+{^5}\text{C}_{\text{4}}+{ ^5}\text{C}_{\text{5}}$ $=\frac{5!}{1!4!}+\frac{5!}{2!3!}+\frac{5!}{3!2!}+\frac{5!}{4!1!}+\frac{5!}{5!0!}$ $=5+\frac{5\times4}{2}+\frac{5\times4}{2}+5+1$ $=5+10+10+5+1$ $=31$

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