The smallest natural number n, such that the coefficient of x in the expansion of $${\left( {{x^2} + {1 \over {{x^3}}}} \right)^n}$$ is ⁿC₂₃, is :

If the coefficients of x² and x³ are both zero, in the expansion of the expression (1 + ax + bx² ) (1 – 3x)¹⁵ in powers of x, then the ordered pair (a,b) is equal to :

If some three consecutive in the binomial expansion of (x + 1)ⁿ is powers of x are in the ratio 2 : 15 : 70, then the average of these three coefficient is :-

If the fourth term in the binomial expansion of $${\left( {{2 \over x} + {x^{{{\log }_8}x}}} \right)^6}$$ (x > 0) is 20 × 8⁷, then a value of x is :