The sum of coefficients in $(1 + x - 3x^2)^{2134}$ is

Let $(a + bx + cx^2)^{10} = \sum_{i=0}^{20} p_i x^i$,where $a, b, c \in N$. If $p_1 = 20$ and $p_2 = 210$,then $2(a + b + c)$ is equal to

Using the Binomial Theorem,evaluate $(99)^{5}$.

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x+\sqrt{x^3-1})^5+(x-\sqrt{x^3-1})^5, x>1$. If $u$ and $v$ satisfy the equations $\alpha u+\beta v=18$ and $\gamma u+\delta v=20$,then $u+v$ equals:

Find the expansion of $(3 x^{2}-2 a x+3 a^{2})^{3}$ using the binomial theorem.

Prove that $\sum\limits_{r = 0}^n {3^r \,^nC_r} = 4^n$.