Binomial Theore...

In the expansion of $$(5^{\tfrac 12} + 2^{\tfrac 18})^{1024}$$, the number of integral terms is

if the coefficient of the middle term in the expansion of$$\displaystyle (1+x)^{2n+2}$$and $$p$$ and the coefficients of middle terms in the expansion of$$\left ( 1+x \right )^{2n+1}$$are $$q$$ and $$r$$,then

If the $$(n+1)$$ numbers $$a,b,c,d,...$$ be all different and each of them a prime number, then the number of different factors (other than 1) of $$a^m.b.c.d....$$ is

The total number of terms which are dependent on the value of $$x$$ in the expansion of $$\left(x^2 - 2 + \displaystyle\frac{1}{x^2}\right)^n$$ is equal to   

$$\cfrac { { C }_{ 0 } }{ x } -\cfrac { { C }_{ 1 } }{ x+1 } +\cfrac { { C }_{ 2 } }{ x+2 } -......+{ \left( -1 \right)  }^{ n }\cfrac { { C }_{ n } }{ x+n } =$$_______ where $${ C }_{ r }$$ stands for $${ _{  }^{ n }{ C } }_{ r }$$.

If $$c_{0},c_{1},c_{2}\cdots c_{n}$$ are binomial coefficients in $$\left ( 1+x \right )^{n}$$, then the value of $$c_{1} + c_{5} + c_{9}+c_{13}+\cdots $$ equals

The expresion

The coefficient of $$x^{15}$$ in the product
$$\left ( 1-x \right )\left ( 1-2x \right )\left ( 1-2^{2}.x \right )\left ( 1-2^{3}.x \right )...\left ( 1-2^{15}.x \right )$$
is equal to

The coefficient of $$x^{n-2}$$ in the polynomial $$(x-1)(x-2)(x-3)....(x-n)$$ is 

If $$C_{0},C_{1},C_{2}....,C_{n}$$ denote the binomial coefficients in the expansion of $$\left ( 1+x \right )^{n}$$, then $$\cfrac{C1}{C0}+2\cfrac{C2}{C1}++3\cfrac{C3}{C2}+.....+n\cfrac{Cn}{Cn-1}$$ equals