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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