Binomial Theore...

If the fourth term in the expansion of $$\displaystyle { \left( \sqrt { \frac { 1 }{ { x }^{ \log { x } +1 } }  } +{ x }^{ 1/12 } \right)  }^{ 6 }$$ is equal to $$200$$ and $$x>1$$, then $$x$$ is equal to

If $$A$$ and $$B$$ are coefficients of $${x}^{n}$$ in the expansions of $${ (1+x) }^{ 2n }$$ and $${ (1+x) }^{ 2n-1 }$$ respectively, then $$\dfrac AB$$ is equal to

The expression $${ C }_{ 0 }+2{ C }_{ 1 }+3{ C }_{ 2 }+......+(n+1){ C }_{ n }$$ is equal to

The number of irrational terms in the expansion of $${ ({ 5 }^{\tfrac 16 }+{ 2 }^{ \tfrac 18 }) }^{ 100 }$$ is

If $$f(n)=\sum _{ k=1 }^{ n }{ \sum _{ j=k }^{ n }{ ({ _{  }^{ n }{ C } }_{ j } } ) } ({ _{  }^{ j }{ C } }_{ k })$$, find $$f(n)$$.

Value of $$S=1\times 2\times 3\times 4+2\times 3\times 4\times 5+.......+n(n-1)(n+2)(n+3)$$ is

If the third term in the expansion of
$${ \left( \cfrac { 1 }{ x } +{ x }^{ \log _{ 2 }{ x }  } \right)  }^{ 5 }$$ is $$40$$ then $$x$$ equals

If the coefficients of 2nd, 3rd and the 4th terms in the expansion of $${ \left( 1+x \right)  }^{ n }$$ are in A.P, then value of $$n$$ is

If the third term in the expansion $${ \left( x+{ x }^{ \log _{ 5 }{ x }  } \right)  }^{ 5 }$$ is $$2$$, then $$x$$ equals

Value of
$$\sum _{ k=1 }^{ \infty  } \sum _{ r=0 }^{ k }{ \cfrac { 1 }{ { 3 }^{ k } }  } ({ _{  }^{ k }{ C } }_{ r })$$ is