Question

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

• A. $2^{16}-1$
• B. $2^{13}-14$
• C. $2^{14}$
• D. $2^{13}-13$

Answer 1

Answer: (B)

$2^{13}-14$

Answer 2

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