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Statement $-1$: $\mathop \sum \limits_{r = 0}^n \left( {r + 1} \right)\left( {\begin{array}{*{20}{c}}n\\r\end{array}} \right) = \left( {n + 2} \right){2^{n - 1}}$
Statement $-2$:$\;\mathop \sum \limits_{r = 0}^n \left( {r + 1} \right)\left( {\begin{array}{*{20}{c}}n\\r\end{array}} \right){x^r}\; = {\left( {1 + x} \right)^n} + nx{\left( {1 + x} \right)^{n - 1}}$
Statement $-1$ is false, Statement$-2$ is true
Statement $-1$ is true Statement$-2$ is false;
Statement $-1$ is true, Statement$-2$ is true; Statement $-2$ is not a correct explanation for Statement $-1$
Statement $-1$ is true, Statement$-2$ is true; Statement $-2$ is a correct explanation for Statement $-1$
Solution
We have
$\sum_{r=0}^{n}(r+1)^{n} C_{r} x^{r}=$$\sum_{r=0}^{n} r {\cdot}^{n} C_{r} x^{r}+\sum_{r=0}^{n}{\cdot} ^{n} C_{r} x^{r}$
$=\sum_{r=1}^{n} r \cdot \frac{n}{r}^{n-1} C_{r-1} x^{r}+(1+x)^{n}$
$=n x \sum_{r=1}^{n} n-^{1} C_{r-1} x^{r-1}+(1+x)^{n}$
$=n x(1+x)^{n-1}+(1+x)^{n}=R H S$
$\therefore$ Statement $2$ is correct.
Putting $x=1,$ we get
$\sum_{r=0}^{n}(r+1)^{n} C_{r}=n .2^{n-1}+2^{n}=(n+2) \cdot 2^{n-1}$
Statement $1$ is also true and statement $2$ is a correct explanation for statement $1$
