Binomial Theorem for Positive Integral Index

If in the binomial expansion of $(1-x{)}^m(1+x{)}^n$, the coefficients of $x$ and $x^2$ are respectively $3$ and $-4$, then the ratio $m: n$ is equal to

In the expansion of $(1+x{)}^n$, what is the sum of even index Binomial coefficients?

If ${\left(1+x-2x^2\right)}^6=1+a_{1}x+a_2{x}^2+\dots +a_{12}{x}^{12},$ then
consider the following statements.
$\mathrm{I}.$ $a_2+a_4+a_6+\dots +a_{12}=31$
$\mathrm{II}.$ $a_1+a_3+a_5+\dots +a_{11}=-31$

Which of the above statement(s) is/are correct?

If $p=(8+3\sqrt{7}{)}^n$ and $f=p-\left[p\right],$ where $[·]$ denotes the greatest integer function, then the value of $p(1-f)$ is equal to

The fractional part of $\frac{2^{4n}}{15}$ is $(n\in N)$

The term independent of $a$ in the expansion of ${\left(1+\sqrt{a}+\frac{1}{\sqrt{a}-1}\right)}^{-30}$ is

The remainder, when ${27}^{10}+7^{51}$ is divided by $10$, is

The sum of last three digits of $3^{100}$ is

Let ${\left(5+2\sqrt{6}\right)}^n=p+f$, where $n\in N,p\in N$ and $0<f<1$, then the value of $f^2-f+pf-p$ is

The coefficient of $x^{301}$ in the expansion of $(1+x{)}^{500}+x(1+x{)}^{499}+x^2(1+x{)}^{498}+\dots +x^{500}$ is

The binomial coefficient series $2C_0+\frac{2^2}{2}{C}_1+\frac{2^3}{3}{C}_2+\frac{2^4}{4}{C}_3+\dots \dots +\frac{2^{11}}{11}{C}_{10}$, is