If the coefficents of ${x^3}$ and ${x^4}$ in the expansion of $\left( {1 + ax + b{x^2}} \right){\left( {1 - 2x} \right)^{18}}$ in powers of $x$ are both zero, then $ (a,b) $ is equal to
($14$,$\frac{{272}}{3}$)
($16$,$\frac{{272}}{3}$)
($16$,$\frac{{251}}{3}$)
($14$,$\frac{{251}}{3}$)
The natural number $m$, for which the coefficient of $x$ in the binomial expansion of $\left( x ^{ m }+\frac{1}{ x ^{2}}\right)^{22}$ is $1540,$ is
If sum of the coefficient of the first, second and third terms of the expansion of ${\left( {{x^2} + \frac{1}{x}} \right)^m}$ is $46$, then the coefficient of the term that doesnot contain $x$ is :-
In the expansion of ${\left( {x + \frac{2}{{{x^2}}}} \right)^{15}}$, the term independent of $x$ is
Let the coefficients of three consecutive terms in the binomial expansion of $(1+2 x)^{ n }$ be in the ratio $2: 5: 8$. Then the coefficient of the term, which is in the middle of these three terms, is $...........$.
The interval in which $x$ must lie so that the greatest term in the expansion of ${(1 + x)^{2n}}$ has the greatest coefficient, is