The value of r for which ²⁰C_r ²⁰C₀ + ²⁰C_r$$-$$1 ²⁰C₁ + ²⁰C_r$$-$$2 ²⁰C₂ + . . . . .+ ²⁰C₀ ²⁰C_r  is maximum, is

The sum of the real values of x for which the middle term in the binomial expansion of $${\left( {{{{x^3}} \over 3} + {3 \over x}} \right)^8}$$ equals 5670 is :

The positive value of $$\lambda $$ for which the co-efficient of x² in the expression x² $${\left( {\sqrt x + {\lambda \over {{x^2}}}} \right)^{10}}$$ is 720, is -

If  $${\sum\limits_{i = 1}^{20} {\left( {{{{}^{20}{C_{i - 1}}} \over {{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3} = {k \over {21}}$$  then k is equal to