If $${{}^{20}{C_r}}$$ is the co-efficient of x^r in the expansion of (1 + x)²⁰, then the value of $$\sum\limits_{r = 0}^{20} {{r^2}.{}^{20}{C_r}} $$ is equal to :

A possible value of 'x', for which the ninth term in the expansion of $${\left\{ {{3^{{{\log }_3}\sqrt {{{25}^{x - 1}} + 7} }} + {3^{\left( { - {1 \over 8}} \right){{\log }_3}({5^{x - 1}} + 1)}}} \right\}^{10}}$$ in the increasing powers of $${3^{\left( { - {1 \over 8}} \right){{\log }_3}({5^{x - 1}} + 1)}}$$ is equal to 180, is :

If the coefficients of x⁷ in $${\left( {{x^2} + {1 \over {bx}}} \right)^{11}}$$ and x^$$-$$7 in $${\left( {{x} - {1 \over {bx^2}}} \right)^{11}}$$, b $$\ne$$ 0, are equal, then the value of b is equal to :

The sum of all those terms which are rational numbers in the

expansion of (2^1/3 + 3^1/4)¹² is :