The value of
$-{ }^{15} \mathrm{C}_{1}+2 \cdot{ }^{15} \mathrm{C}_{2}-3{ }^{15} \mathrm{C}_{3}+\ldots . .-15 \cdot{ }^{15} \mathrm{C}_{15}+{ }^{14} \mathrm{C}_{1}+{ }^{14} \mathrm{C}_{3}+{ }^{14} \mathrm{C}_{5}+\ldots .+{ }^{14} \mathrm{C}_{11}$ is:
Correct Option: , 4
$\mathrm{S}_{1}=-{ }^{15} \mathrm{C}_{1}+2 \cdot{ }^{15} \mathrm{C}_{2}-\ldots \ldots \ldots . . .15^{15} \mathrm{C}_{15}$
$=\sum_{r=1}^{15}(-1)^{r} \cdot r^{15} C_{r}=15 \sum_{r-1}^{15}(-1)^{r 14} C_{r-1}$
$=15\left(-{ }^{14} C_{0}+{ }^{14} C_{1}-\ldots \ldots-{ }^{14} C_{14}\right)=15(0)=0$
$\mathrm{S}_{2}={ }^{14} \mathrm{C}_{1}+{ }^{14} \mathrm{C}_{3}+\ldots \ldots+{ }^{14} \mathrm{C}_{11}$
$=\left({ }^{14} \mathrm{C}_{1}+{ }^{14} \mathrm{C}_{3}+\ldots \ldots+{ }^{14} \mathrm{C}_{11}+{ }^{14} \mathrm{C}_{13}\right)-{ }^{14} \mathrm{C}_{13}$
$=2^{13}-14$
$=\mathrm{S}_{1}+\mathrm{S}_{2}=2^{13}-14$
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