Binomial Theore...

If $${ C }_{ 0 },{ C }_{ 1 },{ C }_{ 2 },...,{ C }_{ n }$$ are the coefficients of the expansion of $${ \left( 1+x \right)  }^{ n }$$, then the value of $$\displaystyle \sum _{ 0 }^{ n }{ \frac { { C }_{ k } }{ k+1 }  } $$ is

If $$n$$ is an integer between $$0$$ and $$21$$, then the  minimum value of $$n!(21 - n)!$$ is attained for $$n=$$

If the coefficient of $$x^7$$ in $$\left[ax + \left(\displaystyle\frac{1}{bx}\right)\right]^{11} is\ 55 a^{11}$$, then $$a$$ and $$b$$ satisfy the relation   

If $${ \left( 1+x \right)  }^{ n }={ C }_{ 0 }+{ C }_{ 1 }x+{ C }_{ 2 }{ x }^{ 2 }+...+{ C }_{ n }{ x }^{ n }$$, then $$\displaystyle 2{ C }_{ 0 }+{ 2 }^{ 2 }.\frac { { C }_{ 1 } }{ 2 } +{ 2 }^{ 3 }.\frac { { C }_{ 2 } }{ 3 } +...+{ 2 }^{ n+1 }.\frac { { C }_{ n } }{ n+1 } =$$

The value of $$\displaystyle \frac { { _{  }^{ n }{ C } }_{ 1 }^{  } }{ 2 } +\frac { { _{  }^{ n }{ C } }_{ 3 }^{  } }{ 4 } +\frac { { _{  }^{ n }{ C } }_{ 5 }^{  } }{ 6 } +...$$ is

Find the sum 1 $$\times$$ 2 $$\times$$ C$$_1$$ + 2 $$\times$$ 3 C$$_2$$ + n (n+1)C$$_{n'}$$ where C$$_r$$ = $$^n$$C$$_r$$. 

If$$(1+2x+x^2)^n = \displaystyle \sum_{r=0}^{2n} a_r x^r$$, then $$a_r=$$

Find the ratio of the coefficient of $${ x }^{ 10 }$$ in $${ \left( 1-{ x }^{ 2 } \right)  }^{ 10 }$$ and the term independent of $$x$$ in the expansion of $${ \left( x-\cfrac { 2 }{ x }  \right)  }^{ 10 }$$

The sum of the coefficients in the expansion of $${ \left( 1+5x-7{ x }^{ 3 } \right)  }^{ 3165 }$$ is

Which term in the expansion of $${ \left[ \sqrt [ 3 ]{ \left( \frac { a }{ \sqrt { b }  }  \right)  } +\sqrt { \left( \frac { b }{ \sqrt [ 3 ]{ a }  }  \right)  }  \right]  }^{ 21 }$$ contains $$a$$ and $$b$$ to one and same power.