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If for positive integers $r > 1$ and $n > 2$,the coefficients of the $(3r)^{th}$ and $(r + 2)^{th}$ powers of $x$ in the expansion of $(1 + x)^{2n}$ are equal,then:

The numerically greatest term in the expansion of $(x+3y)^{13}$,when $x=\frac{1}{2}$ and $y=\frac{1}{3}$ is

Number of rational terms in the expansion of $( \sqrt{2} + \sqrt[4]{3} )^{100}$ is:

If the term independent of $x$ in the expansion of $(\sqrt{a}x^2 + \frac{1}{2x^3})^{10}$ is $105$,then $a^2$ is equal to:

In the binomial $(2^{1/3} + 3^{-1/3})^n$,if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is $1/6$,then $n =$

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