Question:
If ${ }^{20} \mathrm{C}_{\mathrm{r}}$ is the co-efficient of $\mathrm{x}^{\mathrm{r}}$ in the expansion of $(1+x)^{20}$, then the value of $\sum_{r=0}^{20} r^{2}{ }^{20} C_{r}$ is equal to $:$
Correct Option: 4,
Solution:
$\sum_{r=0}^{20} r^{2} \cdot{ }^{20} C_{r}$
$\sum\left(4(\mathrm{r}-1+) \mathrm{r}{ }^{20} \mathrm{C}_{\mathrm{r}}\right)$
$\sum r\left(r-1 \cdot \frac{20 \times 19}{r(r-1)} \cdot{ }^{18} C_{r}+r \cdot \frac{20}{r} \cdot \sum{ }^{19} C_{r-1}\right.$
$\Rightarrow 20 \times 19.2^{18}+20.2^{19}$
$\Rightarrow 420 \times 2^{18}$
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