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  4. If the coefficient of x2 r is greater than half of the coefficient of x2 r+1 in the expansion of (1+x)15 then the possible value of r is -

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Q. If the coefficient of $x^{2 r}$ is greater than half of the coefficient of $x^{2 r+1}$ in the expansion of $(1+x)^{15}$ then the possible value of $r$ is -

Binomial Theorem

Solution:

${ }^{15} C _{2 r }>\frac{1}{2}{ }^{15} C _{2 r +1}$
$2>\frac{{ }^{15} C _{2 r +1}}{{ }^{15} C _{2 r }} \Rightarrow 2>\frac{15-(2 r +1)+1}{2 r +1} \Rightarrow 4 r +2>15-2 r \Rightarrow 6 r >13 \Rightarrow r >\frac{13}{6}$
& $2 r +1 \leq 15 \Rightarrow 2 r \leq 14 \Rightarrow r \leq 7$
$\therefore$ Options A, B & C are correct.