${n^n}{\left( {\frac{{n + 1}}{2}} \right)^{2n}}$ is
Less than ${\left( {\frac{{n + 1}}{2}} \right)^3}$
Greater than ${\left( {\frac{{n + 1}}{2}} \right)^3}$
Greater than ${(n!)^3}$
$(b)$ and $(c)$ both
If the expansion in powers of $x$ of the function $\frac{1}{{\left( {1 - ax} \right)\left( {1 - bx} \right)}}$ is ${a_0} + {a_1}x + {a_2}{x^2} + \;{a_3}{x^3} + \; \ldots......$ then ${a_n}$ is
If $\mathrm{b}$ is very small as compared to the value of $\mathrm{a}$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b}+\frac{1}{a-2 b}+\frac{1}{a-3 b} \ldots .+\frac{1}{a-n b}=\alpha n+\beta n^{2}+\gamma n^{3}$, then the value of $\gamma$ is:
Coefficient of $x^{19}$ in the polynomial $(x-1) (x-2^1) (x-2^2) .... (x-2^{19})$ is
If ${(1 - x + {x^2})^n} = {a_0} + {a_1}x + {a_2}{x^2} + .... + {a_{2n}}{x^{2n}}$, then ${a_0} + {a_2} + {a_4} + .... + {a_{2n}} = $
In the expansion of ${(1 + x)^n}$ the sum of coefficients of odd powers of $x$ is