Let $(1 + x)(1 + x + x^2)(1 + x + x^2 + x^3)\,\, ......\,\,$$(1 + x + x^2 + ..... + x^{30}) = $$a_0 + a_1x + a_2x^2$ .....$+$ $a_{465}x^{465}$, then sum of $a_0 + a_2 + a_4 + ......... +$ is

Co-efficient of $\alpha ^t$ in the expansion of,

$(\alpha + p)^{m - 1} + (\alpha + p)^{m - 2} (\alpha + q) + (\alpha + p)^{m - 3} (\alpha + q)^2 + ...... (\alpha + q)^{m - 1}$

where $\alpha \ne - q$ and $p \ne q$ is :

Total number of terms in the expansion of $\left[ {{{\left( {1 + x} \right)}^{100}} + {{\left( {1 + {x^2}} \right)}^{100}}{{\left( {1 + {x^3}} \right)}^{100}}} \right]$ is

Let $a =$ Minimum $\{x^2 + 2x + 3, x \in R\}$ and $b = \mathop {\lim }\limits_{\theta  \to 0} \frac{{1 - \cos \theta }}{{{\theta ^2}}}$ The value of $\sum\limits_{r = 0}^n {{a^r}.{b^{n - r}}} $ is

If the sum of the coefficients in the expansion of $(x+y)^{n}$ is $4096,$ then the greatest coefficient in the expansion is .... .

In the expansion of ${(x + a)^n}$, the sum of odd terms is $P$ and sum of even terms is $Q$, then the value of $({P^2} - {Q^2})$ will be