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  4. 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

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Q. 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

JEE MainJEE Main 2021Binomial Theorem

Solution:

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