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7.Binomial Theorem
hard
$-{ }^{15} C _{1}+2 \cdot{ }^{15} C _{2}-3 \cdot{ }^{15} C _{3}+\ldots .-15 \cdot{ }^{15} C _{15}+{ }^{14} C _{1}+$ ${ }^{14} C _{3}+{ }^{14} C _{5}+\ldots .+{ }^{14} C _{11}$ का मान है
A
$2^{16}-1$
B
$2^{13}-14$
C
$2^{14}$
D
$2^{13}-13$
(JEE MAIN-2021)
Solution
$\left(-{ }^{15} C _{1}+2 .{ }^{15} C _{2}-3 .{ }^{15} C _{3}+\ldots \ldots-15 .{ }^{15} C _{15}\right)$ $+\left({ }^{14} C _{1}+{ }^{14} C _{3}+\ldots .+{ }^{14} C _{11}\right)$
$=\sum_{r=1}^{15}(-1)^{r} \cdot r^{15} C_{r}+\left({ }^{14} C_{1}+{ }^{14} C_{3}+\ldots+{ }^{14} C_{11}+{ }^{14} C_{13}\right)-{ }^{14} C_{3}$
$=\sum_{r=1}^{15}(-1)^{r} 15 \cdot{ }^{14} C_{r-1}+2^{13}-14$
$=15\left(-14 C_{0}+{ }^{14} C_{1} \ldots \ldots . .-14 C_{14}\right)+2^{13}-14$
$=2^{13}-14$
Std 11
Mathematics
