The value of
Question:

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:

  1. (1) $2^{14}$

  2. (2) $2^{13}-13$

  3. (3) $2^{16}-1$

  4. (4) $2^{13}-14$


Correct Option: , 4

Solution:

$\mathrm{S}_{1}=-{ }^{15} \mathrm{C}_{1}+2 \cdot{ }^{15} \mathrm{C}_{2}-\ldots \ldots \ldots . . .15^{15} \mathrm{C}_{15}$

$=\sum_{r=1}^{15}(-1)^{r} \cdot r^{15} C_{r}=15 \sum_{r-1}^{15}(-1)^{r 14} C_{r-1}$

$=15\left(-{ }^{14} C_{0}+{ }^{14} C_{1}-\ldots \ldots-{ }^{14} C_{14}\right)=15(0)=0$

$\mathrm{S}_{2}={ }^{14} \mathrm{C}_{1}+{ }^{14} \mathrm{C}_{3}+\ldots \ldots+{ }^{14} \mathrm{C}_{11}$

$=\left({ }^{14} \mathrm{C}_{1}+{ }^{14} \mathrm{C}_{3}+\ldots \ldots+{ }^{14} \mathrm{C}_{11}+{ }^{14} \mathrm{C}_{13}\right)-{ }^{14} \mathrm{C}_{13}$

$=2^{13}-14$

$=\mathrm{S}_{1}+\mathrm{S}_{2}=2^{13}-14$

 

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