The coefficient of $x^{1012}$ in the expansion of ${\left( {1 + {x^n} + {x^{253}}} \right)^{10}}$ , (where $n \leq 22$ is any positive integer), is
$1$
$^{10}{C_4}$
$4n$
$^{253}{C_4}$
Coefficient of $x^6$ in the binomial expansion ${\left( {\frac{{4{x^2}}}{3}\; - \;\frac{3}{{2x}}} \right)^9}$ is
Coefficient of ${t^{12}}$ in ${\left( {1 + {t^2}} \right)^6}\left( {1 + {t^6}} \right)\left( {1 + {t^{12}}} \right)$ is-
The sum of all those terms which are rational numbers in the expansion of $\left(2^{1 / 3}+3^{1 / 4}\right)^{12}$ is:
Coefficient of $x^3$ in the expansion of $(x^2 - x + 1)^{10} (x^2 + 1 )^{15}$ is equal to
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