Binomial Theore...

Find the sum $$\displaystyle\sum _{ r=1 }^{ n }{ \cfrac { { r }^{ n }{ C }_{ r } }{ { C }_{ r-1 } }  } $$

The middle term in the expansion of $$\displaystyle \left ( 1-\frac{1}{x} \right )^{n}\left ( 1-x \right )^{n},$$ is

If A is the coefficient of the middle term in the expansion of $$\displaystyle (1+x)^{2n}$$ and B and C are the coefficients of two middle terms in the expansion of $$\displaystyle(1+x)^{2n-1}$$, then

The middle term in the expansion of$$\displaystyle (1+x)^{2n}$$is

If the coeffecient of the middle term in the expansion of $${ (1+x) }^{ 2n+2 }$$ is $$\alpha $$ and coeffecient of middle terms in the expansion of $${ (1+x) }^{ 2n+1 }$$ are $$\beta $$ and $$\gamma $$, then relate $$\alpha, \beta$$ and $$ \gamma $$

The middle term in the expansion of $$\displaystyle \left ( x+\frac{1}{x} \right )^{10}$$,is

The middle term in the expansion of $${ (1+x) }^{ 2n }$$ is

In the second term in the expansion $$\displaystyle { \left( \sqrt [ 13 ]{ a } +\frac { a }{ \sqrt { { a }^{ -1 } }  }  \right)  }^{ n }$$ is $$\displaystyle 14{ a }^{ \frac { 5 }{ 2 }  }$$ , then the value of $$\displaystyle \frac { _{  }^{ n }{ { C }_{ 3 }^{  } } }{ ^{ n }{ { C }_{ 2 }^{  } } } $$ is

The coeffecients of the middle term in the binomial expansion in powers of $$x$$ of $${ (1+\alpha x) }^{ 4 }$$ and $${ (1+\alpha x) }^{ 6 }$$ is the same if $$\alpha $$ equals

The coefficient of 1/x in the expansion of $$(1 + x)^n (1 + 1/x)^n$$ is