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Q. The value of $\displaystyle\sum_{r=0}^{s} \displaystyle\sum_{s=1}^{n}{ }^{n} C_{s}{}^{s} C_{r}$ is

Binomial Theorem

Solution:

$\underset{r \le s}{\displaystyle\sum_{r=0}^{s} \displaystyle\sum_{s=1}^{n}}{ }^{n} C_{s}{}^{s} C_{r}$
$=\displaystyle\sum_{s=1}^{n}{ }^{n} C_{s}\left({ }^{s} C_{0}+{ }^{s} C_{1}+{ }^{s} C_{2}+\ldots+{ }^{s} C_{s}\right)$
$=\displaystyle\sum_{s=1}^{n}{ }^{n} C_{s} 2^{s}=\displaystyle\sum_{s=0}^{n}{ }^{n} C_{s} 2^{s}-{ }^{n} C_{0} 2^{0}$
$=(1+2)^{n}-1=3^{n}-1$