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Q. The value of $ \displaystyle\sum_{r=0}^{10} r^{10} C_{r} 3^{r}(-2)^{10-r}$ is

Binomial Theorem

Solution:

$ \displaystyle\sum_{r=0}^{10} r^{10} C_{r} 3^{r}(-2)^{10-r}$
$=10 \displaystyle \sum_{r=0}^{10}{ }^{9} C_{r-1} 3^{r}(-2)^{10-r}$
$=10 \times 3 \displaystyle\sum_{r=0}^{10}{ }^{9} C_{r-1} 3^{r-1}(-2)^{10-r}$
$=30(3-2)^{10}$
$=30$