Let n$$\in$$N and [x] denote the greatest integer less than or equal to x. If the sum of (n + 1) terms $${}^n{C_0},3.{}^n{C_1},5.{}^n{C_2},7.{}^n{C_3},.....$$ is equal to 2¹⁰⁰ . 101, then $$2\left[ {{{n - 1} \over 2}} \right]$$ is equal to _______________.

If the co-efficient of x⁷ and x⁸ in the expansion of $${\left( {2 + {x \over 3}} \right)^n}$$ are equal, then the value of n is equal to _____________.

The ratio of the coefficient of the middle term in the expansion of (1 + x)²⁰ and the sum of the coefficients of two middle terms in expansion of (1 + x)¹⁹ is _____________.

The term independent of 'x' in the expansion of
$${\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}}$$, where x $$\ne$$ 0, 1 is equal to ______________.