The value of -{ }^{15} C _{1}+2 .{ }^{15} C _ | vedclass

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Question

$\left(-{ }^{15} C _{1}+2 .{ }^{15} C _{2}-3 .{ }^{15} C _{3}+\ldots \ldots-15 .{ }^{15} C _{15}\right)$ $+\left({ }^{14} C _{1}+{ }^{14} C _{3}+\ldot

Vedclass · Accepted Answer

$2^{13}-14$

Vedclass · Answer

$\left(-{ }^{15} C _{1}+2 .{ }^{15} C _{2}-3 .{ }^{15} C _{3}+\ldots \ldots-15 .{ }^{15} C _{15}\right)$ $+\left({ }^{14} C _{1}+{ }^{14} C _{3}+\ldots .+{ }^{14} C _{11}\right)$

$=\sum_{r=1}^{15}(-1)^{r} \cdot r^{15} C_{r}+\left({ }^{14} C_{1}+{ }^{14} C_{3}+\ldots+{ }^{14} C_{11}+{ }^{14} C_{13}\right)-{ }^{14} C_{3}$

$=\sum_{r=1}^{15}(-1)^{r} 15 \cdot{ }^{14} C_{r-1}+2^{13}-14$

$=15\left(-14 C_{0}+{ }^{14} C_{1} \ldots \ldots . .-14 C_{14}\right)+2^{13}-14$

$=2^{13}-14$

The value of $-{ }^{15} C _{1}+2 .{ }^{15} C _{2}-3 .{ }^{15} C _{3}+\ldots \ldots$ $-15 .{ }^{15} C _{15}+{ }^{14} C _{1}+{ }^{14} C _{3}+{ }^{14} C _{5}+\ldots .+{ }^{14} C _{11}$ is

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