If n > 1 is an integer and x $$\ne$$ 0, then (1 + x)ⁿ $$-$$ nx $$-$$ 1 is divisible by

The term independent of $x$ in the expansion of $\left(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}}\right)^{15}$ is equal to

If $\sum_{r=0}^{2 n} a_r(x-2)^r=\sum_{r=0}^{2 n} b_r(x-3)^r$ and $a_k=1 \forall k \geq 1$, then the value of $\frac{b_n}{{ }^{2 n+1} C_{n+1}}$ is

If $\left(1+x-2 x^2\right)^6=1+a_1 x+a_2 x^2+\ldots+a_{12} x^{12}$, then the value of $a_2+a_4+a_6+\ldots+a_{12}$ is