If the number of terms in the binomial expansion of $(2x + 3)^{3n}$ is $22$,then the value of $n$ is:

If $b$ is very small as compared to the value of $a$,so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b}+\frac{1}{a-2b}+\frac{1}{a-3b}+\ldots+\frac{1}{a-nb}=\alpha n+\beta n^2+\gamma n^3$,then the value of $\gamma$ is:

The total number of terms in the expansion of $(x+y)^{100} + (x-y)^{100}$ after simplification 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

$(\sqrt{2} + 1)^6 - (\sqrt{2} - 1)^6 = $

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