Binomial Theore...

$$\sum { { \left( -1 \right)  }^{ r } } ~ { _{  }^{ n }{ C } }_{ r }\cfrac { 1+r\log _{ e }{ 10 }  }{ { \left( 1+\log _{ e }{ { 10 }^{ n } }  \right)  }^{ r } } $$

If $$\left\{ x \right\}$$  denotes the fraction part of $$'x'$$, then $$\left\{ \dfrac { { 3 }^{ 1001 } }{ 82 }  \right\} =$$

The coefficient of $$x^{160}$$ in the expansion of $$\displaystyle (x^8 + 1)^{60} \left( x^{12} + 3x^4 + \frac{3}{x^4} + \frac{1}{x^{12}} \right)^{-10}$$ is

If $$C_r \, = \, (^{100 C _ r }) , then \,  E = \sum_{r = 0 }^{n + 4 } (-1)^r  \, c_r \, c_{r + 1}$$

The coefficient of $${x^6}.{y^{ - 2}}$$ in the expansion of $${\left( {\dfrac{{{x^2}}}{y} - \dfrac{y}{x}} \right)^{12}}$$ is

The co-efficient of $${x^{53}}$$ in the expression $$\sum\limits_{m = 0}^{100} {{}^{100}} {c_m}{(x - 3)^{100 - m}}{2^m}\,$$ is

If $${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}\,\,$$ then the sum of the series $$\,1 + {}^n{C_r} + {}^n + 1{C_r} + {}^{n + 2}{C_3} + ......... + {}^{n + r - 1}{C_r}$$ is 

In the expression of $$\left( {{2^x} + \frac{1}{{{4^x}}}} \right)^n\,$$ ratio  of 2nd and third terms is given by$$\,{t_3}/{t_2} = 7$$ and the sum of the co-efficients of 2nd and 3rd term is $$36,$$ then the value of $$x$$ is 

The coefficient $${x^n}$$ in the expression of $${\left( {1 + x} \right)^{2n}}$$ and $${\left( {1 + x} \right)^{2n - 1}}$$ are in the ratio.

The coefficient of $$x^{8}$$ in the polynomial $$\left( x-1 \right) \left( x-2 \right) \left( x-3 \right) ...\left( x-10 \right) $$ is