Binomial Theore...

If the fourth term of $${ \left( \sqrt { { x }^{ \left( \cfrac { 1 }{ 1+\log { x }  }  \right)  } } +\sqrt [ 12 ]{ x }  \right)  }^{ 6 }$$ is equal to 200 and $$x>1$$, then $$x$$ is equal to

The number of irrational terms in the expansion of $$(\sqrt[8]{5}+\sqrt[6]{2})^{100}$$ is

Value of $$P=\sum _{ 0\le  }^{  }{ \sum _{ r<s\le n }^{  }{ { C }_{ r } } { C }_{ s } } $$ is

If $$n$$ is even, then value of the expression
$${ C }_{ 0 }-\cfrac { 1 }{ 2 } { C }_{ 1 }^{ 2 }+\cfrac { 1 }{ 3 } { C }_{ 2 }^{ 2 }-.....+\cfrac { { \left( -1 \right)  }^{ n } }{ n+1 } { C }_{ n }^{ 2 }$$
where
$${ C }_{ r }={ _{  }^{ n }{ C } }_{ r }$$ is

The value of $$ \displaystyle B=\sum_{0\leq r\leq s\leq n}^{} $$ $$ \displaystyle \sum \left ( C_{r}-C_{s} \right )^{2} $$ is

The value of the expression $$\displaystyle \frac{1 + 4\:.\:343 + 7\:.\:4 + 2\:.\:3\:.\:49 + 7\:.\:343}{16 + 2^6\:.\:3^1 + 2^{5}\:.\:3^{3} + 2^{6}\: . \: 3^{3} + 2^{4}\:.\:3^{4}}$$ equal

If in the expansion of $${ \left( { x }^{ 3 }-\cfrac { 1 }{ { x }^{ 2 } }  \right)  }^{ n }$$,
$$n\in N$$, sum of coefficient of $${ x }^{ 5 }$$ and $${ x }^{ 10 }$$ is $$0$$, then value of $$n$$ is

If $${ S }_{ n }=1+q+{ q }^{ 2 }+{ q }^{ 3 }+...+{ q }^{ n }$$ and $$\displaystyle { S' }_{ n }=1+\left( \frac { q+1 }{ 2 }  \right) +{ \left( \frac { q+1 }{ 2 }  \right)  }^{ 2 }+...+{ \left( \frac { q+1 }{ 2 }  \right)  }^{ n },q\neq 1$$ then $$^{ n+1 }{ { C }_{ 1 } }+^{ n+1 }{ { C }_{ 2 } }.{ S }_{ 1 }+^{ n+1 }{ { C }_{ 3 } }.{ S }_{ 2 }+...+^{ n+1 }{ { C }_{ n+1 } }.{ S }_{ n }=$$

$$\left( _{  }^{ m }{ { C }_{ 0 }^{  } }+^{ m }{ { C }_{ 1 }^{  } }-^{ m }{ { C }_{ 2 }^{  } }-^{ m }{ { C }_{ 3 }^{  } } \right) +\left( ^{ m }{ { C }_{ 4 }^{  } }+^{ m }{ { C }_{ 5 }^{  } }-^{ m }{ { C }_{ 6 }^{  } }-^{ m }{ { C }_{ 7 }^{  } } \right) +...=0$$ if and only if for some positive integer $$k, m=$$

The value of $$\displaystyle ^{20}C_{0}+ ^{20}C_{1}+^{20}C_{2}+^{20}C_{4}+^{20}C_{12}+^{20}C_{13}+^{20}C_{14}+^{20}C_{15} $$ is