Find an approximation of $(0.99)^{5}$ using the first three terms of its binomial expansion.

The value of $^nC_1 \sum_{r=0}^1 {^1C_r} + ^nC_2 \left( \sum_{r=0}^2 {^2C_r} \right) + ^nC_3 \left( \sum_{r=0}^3 {^3C_r} \right) + \dots + ^nC_n \left( \sum_{r=0}^n {^nC_r} \right)$ is equal to

The sum of the coefficients in the expansion of $\left(1+\frac{x}{2}\right)^{12}$ is

Find $(a+b)^{4}-(a-b)^{4}$. Hence,evaluate $(\sqrt{3}+\sqrt{2})^{4}-(\sqrt{3}-\sqrt{2})^{4}$. (in $\sqrt{6}$)

If the coefficients of $x^2$ and $x^3$ are both zero in the expansion of the expression $(1 + ax + bx^2)(1 - 3x)^{15}$ in powers of $x$,then the ordered pair $(a, b)$ is equal to

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