Binomial Theore...

The $$4th$$ term from the end in the expansion of $${ \left( \cfrac { { x }^{ 3 } }{ 2 } -\cfrac { 2 }{ { x }^{ 2 } }  \right)  }^{ 7 }$$ is

The middle term in the expansion of $${ \left( \cfrac { a }{ x } +bx \right)  }^{ 12 }$$ is

Find the middle term in the expansion of $${ \left( 3x-\cfrac { { x }^{ 3 } }{ 6 }  \right)  }^{ 9 }$$.

There will be no term containing $${ x }^{ 2r }$$ in the expansion of $${ \left( x+{ x }^{ -2 } \right)  }^{ n-3 }$$ if $$(n-2r)$$ is positive but not a multiple of 

If $${ \left( 8+3\sqrt { 7 }  \right)  }^{ n }=\alpha +\beta $$  where $$n$$ and $$\alpha$$ are positive integers and $$\beta$$ is a positive proper fraction,then

If $${ \left( 1+x \right)  }^{ n }={ C }_{ 0 }+{ C }_{ 1 }x+{ C }_{ 2 }{ x }^{ 2 }+...+{ C }_{ n }{ x }^{ n }$$, then $$\displaystyle \sum _{ 0\le i\le  }^{  }{ \sum _{ j\le n }^{  }{ { \left( { C }_{ i }+{ C }_{ j } \right)  }^{ 2 } } = } $$

If the second term in the expansion $${ \left[ a^{\dfrac {1}{13}} +\dfrac { a }{ \sqrt { { a }^{ -1 } }  }  \right]  }^{ n }$$ is $$14\ { a }^{ 5/2 }$$, then the value of $$\dfrac {^{n}C_{3}}{^{n}C_{2}}$$ is

If the number of terms in $${ \left( x+1+\cfrac { 1 }{ x }  \right)  }^{ n }\quad (n\in { I }^{ + })$$ is 401, then $$n$$ is greater than

If $$\displaystyle{ a }_{ n }=\sum _{ r=0 }^{ n }{ \cfrac { 1 }{ { _{  }^{ n }{ C } }_{ r } }  } $$then $$\displaystyle\sum _{ r=0 }^{ n }{ \cfrac { r }{ { _{  }^{ n }{ C } }_{ r } }  }$$ equals

The total number of terms in the expansion of $${ \left( x+a \right)  }^{ 100 }+{ \left( x-a \right)  }^{ 100 }$$  after simplification is