Binomial Theore...

Value of $$\sum { \sum _{ 0\le r<s\le n }^{  }{ { (r+s)\left( { C }_{ r }+{ C }_{ s } \right)  }^{ 2 } }  } $$ is

If $$A=^{ 2n }{ { C }_{ 0 } }.^{ 2n }{ { C }_{ 1 } }+^{ 2n }{ { C }_{ 1 } }^{ 2n-1 }{ { C }_{ 1 } }+^{ 2n }{ { C }_{ 2 } }^{ 2n-2 }{ { C }_{ 1 } }+...$$ then $$A$$ is

The coefficient of $${ x }^{ 9 }$$ in the expansion of $${ \left( { x }^{ 3 }+\cfrac { 1 }{ { 2 }^{ t } }  \right)  }^{ 11 }$$, where $$t=\log _{ \sqrt { 2 }  }{ { (x }^{ \tfrac 32 } } )$$,

Value of $$\sum { \sum _{ 0\le i<j\le n }^{  }{ { \left( { C }_{ i }+{ C }_{ j } \right)  }^{ 2 } }  } $$ is

Value(s) of $$x$$ for which the fourth term in the expansion of
$${ \left( { \sqrt { x }  }^{ 1/(\log _{ 2 }{ x+1 } ) }+{ x }^{ 1/2 } \right)  }^{ 6 }$$ is $$40$$ is (are)

If $${x}^{2r}$$ occurs in $${ \left( x+\cfrac { 2 }{ { x }^{ 2 } }  \right)  }^{ n }$$, then $$n-2r$$ must be of the form

If the middle term of $${ \left( x+\cfrac { 1 }{ x } \sin ^{ -1 }{ x }  \right)  }^{ 8 }$$ is $$\cfrac { 35{ \pi  }^{ 4 } }{ 8 } $$, then value of $$x$$ can be

Value of the expression
$$\quad { C }_{ 0 }+({ C }_{ 0 }+{ C }_{ 1 })+({ C }_{ 0 }+{ C }_{ 1 }+{ C }_{ 2 })+....+({ C }_{ 0 }+{ C }_{ 1 }+....+{ C }_{ n-1 })$$ is

If $${ S }_{ n }=\cfrac { 1 }{ n+1 } \sum _{ i=0 }^{ n }{ \begin{pmatrix} n \\ i \end{pmatrix} } $$, then $$2{ S }_{ n+1 }-{ S }_{ n }$$ equals

The number of irrational terms in the expansion of $${ \left( { 4 }^{ 1/5 }+{ 7 }^{ 1/10 } \right)  }^{ 45 }$$ is