If the number of terms in the expansion of $(x - 2y + 3z)^n$ is $45$,then $n=$

The sum of the coefficients in the expansion of $(1+x+x^2)^n$ is

Expand the expression $\left(\frac{2}{x}-\frac{x}{2}\right)^{5}$

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:

Let $a$ be the largest real root and $b$ be the smallest real root of the polynomial equation $x^6-6x^5+15x^4-20x^3+15x^2-6x+1=0$. Then $\frac{a^2+b^2}{a+b+1}$ is

Using the Binomial Theorem,evaluate $(102)^{5}$.