The value of $-{ }^{15} C_{1}+2 \cdot{ }^{15} C_{2}-3 \cdot{ }^{15} C_{3}+\ldots \ldots$ $-15 .{ }^{15} \mathrm{C}_{15}+{ }^{14} \mathrm{C}_{1}+{ }^{14} \mathrm{C}_{3}+{ }^{14} \mathrm{C}_{5}+\ldots .+{ }^{14} \mathrm{C}_{11}$ is :
Correct Option: , 2
$\left(-{ }^{15} C_{1}+2 .{ }^{15} C_{2}-3 .{ }^{15} C_{3}+\ldots \ldots-15 .{ }^{15} C_{15}\right)$
$+\left({ }^{14} C_{1}+{ }^{14} C_{3}+\ldots .+{ }^{14} C_{11}\right)$
$=\sum_{r=1}^{15}(-1)^{r} \cdot r{ }^{15} C_{r}+\left({ }^{14} C_{1}+{ }^{14} C_{3}+\ldots+{ }^{14} C_{11}+{ }^{14} C_{13}\right)-{ }^{14} C_{3}$
$=\sum_{\mathrm{r}=1}^{15}(-1)^{\mathrm{r}} 15 \cdot{ }^{14} \mathrm{C}_{\mathrm{r}-1}+2^{13}-14$
$=15\left(-{ }^{14} C_{0}+{ }^{14} C_{1} \ldots \ldots . .-{ }^{14} C_{14}\right)+2^{13}-14$
$=2^{13}-14$
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