Binomial Theore...

Let $$n$$ and $$k$$ be  positive integers such that $$\displaystyle n\ge \frac { k\left( k+1 \right)  }{ 2 } .$$ The number of solution $$\left( { x }_{ 1 },{ x }_{ 2 },..,{ x }_{ k } \right) \ge 1;{ x }_{ 2 }\ge 2,...,{ x }_{ k }\ge k$$ all integers satisfying $${ x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 }+...+{ x }_{ k }=n$$ is

The number of irrational terms in the expansion of $${ \left( { 2 }^{ \dfrac 15 }+{ 3 }^{ \dfrac {1}{10} } \right)  }^{ 55 }$$ is

Find the middle term in the expansion of $${ \left( \cfrac { x }{ a } -\cfrac { a }{ x }  \right)  }^{ 21 }$$

In the expansion of the expression $${ \left( x+a \right)  }^{ 15 }$$, if the eleventh term in the geometric mean of the eighth and twelfth terms, which term in the expression is the greatest?

Find the sum of the series $$\displaystyle\sum _{ r=0 }^{ n }{ { \left( -1 \right)  }^{ n } }  { _{  }^{ n }{ C } }_{ r }\left[ \cfrac { 1 }{ { 2 }^{ r } } +\cfrac { { 3 }^{ r } }{ { 2 }^{ 2r } } +\cfrac { { 7 }^{ r } }{ { 2 }^{ 3r } } +\cfrac { { 15 }^{ r } }{ { 2 }^{ 4r } } +...upto\: m\: terms \right] $$

Determine the value of $$x$$ in the expression $${ \left( x+{ x }^{ t } \right)  }^{ 5 }$$, if the third term in the expression is 10,00,000 where $$t=\log _{ 10 }{ x } $$.

If $$\cfrac { { _{  }^{ n }{ C } }_{ r }+4{ _{  }^{ n }{ C } }_{ r+1 }+6{ _{  }^{ n }{ C } }_{ r+2 }+4{ _{  }^{ n }{ C } }_{ r+3 }+{ _{  }^{ n }{ C } }_{ r+4 } }{ { _{  }^{ n }{ C } }_{ r }+3{ _{  }^{ n }{ C } }_{ r+1 }+3{ _{  }^{ n }{ C } }_{ r+2 }+{ _{  }^{ n }{ C } }_{ r+3 } } =\cfrac { n+k }{ r+k } $$. Find the value of k

The value of $$\cfrac { 1 }{ \left( n-1 \right) ! } +\cfrac { 1 }{ \left( n-3 \right) !3! } +\cfrac { 1 }{ \left( n-5 \right) !5! }+.... $$

If $${ \left( 1+x \right)  }^{ n }={ C }_{ 0 }+{ C }_{ 1 }x+{ C }_{ 2 }{ x }^{ 2 }+..........+{ C }_{ n }{ x }^{ R }$$, then the sum
$${ C }_{ 0 }+({ C }_{ 0 }+{ C }_{ 1 })+({ C }_{ 0 }+{ C }_{ 1 }+{ C }_{ 2 })+.....+({ C }_{ 0 }+{ C }_{ 1 }+{ C }_{ 2 }+.....+{ C }_{ n-1 }$$ is

If $$n$$ is a positive integer and $${ C }_{ k }={ _{  }^{ n }{ C } }_{ k }$$, find the value of $$\sum _{ k=1 }^{ n }{ { k }^{ 3 }{ \left( \cfrac { { C }_{ k } }{ { C }_{ k-1 } }  \right)  }^{ 2 } } $$