The coefficient of $x^{50}$ in the binomial expansion of ${\left( {1 + x} \right)^{1000}} + x{\left( {1 + x} \right)^{999}} + {x^2}{\left( {1 + x} \right)^{998}} + ..... + {x^{1000}}$ is
$\frac{{\left( {1000} \right)!}}{{\left( {50} \right)!\left( {950} \right)!}}$
$\frac{{\left( {1000} \right)!}}{{\left( {49} \right)!\left( {951} \right)!}}$
$\frac{{\left( {1001} \right)!}}{{\left( {51} \right)!\left( {950} \right)!}}$
$\frac{{\left( {1001} \right)!}}{{\left( {50} \right)!\left( {951} \right)!}}$
The coefficient of ${x^4}$ in the expansion of ${(1 + x + {x^2} + {x^3})^n}$ is
The coefficient of $t^4$ in the expansion of ${\left( {\frac{{1 - {t^6}}}{{1 - t}}} \right)^3}$ is
The positive value of $a$ so that the co-efficient of $x^5$ is equal to that of $x^{15}$ in the expansion of ${\left( {{x^2}\,\, + \,\,\frac{a}{{{x^3}}}} \right)^{10}}$ is
If the coefficient of $x ^7$ in $\left(a x-\frac{1}{b x^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(a x+\frac{1}{b x^2}\right)^{13}$ are equal, then $a^4 b^4$ is equal to :
In the expansion of ${\left( {3x - \frac{1}{{{x^2}}}} \right)^{10}}$ then $5^{th}$ term from the end is :-