The coefficient of $x^{50}$ in the expansion of $(1+x)^{100}+2x(1+x)^{99}+3x^2(1+x)^{98}+\dots+101x^{100}$ is:

For an integer $n \geq 2$,if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2n-3}$ is $16$,then the distance of the point $P(2n-1, n^2-4n)$ from the line $x+y=8$ is:

If the sum of the coefficients of all even powers of $x$ in the product $(1+x+x^{2}+\ldots+x^{2n})(1-x+x^{2}-x^{3}+\ldots+x^{2n})$ is $61$,then $n$ is equal to

If $C_r$ denotes the binomial coefficient ${ }^{n} C_r$,then $(-1) C_0^2+2 C_1^2+5 C_2^2+\ldots+(3 n-1) C_n^2$ is equal to

The fractional part of a real number $x$ is defined as $x - [x]$,where $[x]$ is the greatest integer less than or equal to $x$. Let $F_1$ and $F_2$ be the fractional parts of $(44 - \sqrt{2017})^{2017}$ and $(44 + \sqrt{2017})^{2017}$,respectively. Then,$F_1 + F_2$ lies between the numbers:

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