Binomial Theore...

The sum of the series $$\displaystyle\sum_{r=0}^{n}\left ( ^{n+1}C_{r} \right ) $$ equals

the $$7th$$ term in $$\displaystyle { \left( \frac { 1 }{ y } +{ y }^{ 2 } \right)  }^{ 10 }$$, when expanded in descending power of $$y$$, is

Number of rational term is the expansion of $$\left ( 7^{1/3}+11^{1/9} \right )^{729}$$

The values of $$x$$ in the expansion $$\displaystyle \left ( x+x^{log_{10}x} \right )^{5}$$ , if the third term in the expansion is $$10,00,000$$

If the sum of the coefficients in the expansion of $$\left ( 1+2x^{} \right )^{m}$$ and $$\left ( 2+x \right )^{n}$$ are respectively $$6561$$ and $$243$$, then the position of the point $$\left ( m,n \right )$$ with respect to circle $$x^{2}+y^{2}-4x-6y-32=0$$

If the sum of the coefficients in the expansion of $$(1+2x)^{n}$$ is $$2,187$$, the greatest  term in the expansion, if $$\displaystyle x = \frac{1}{2}$$ is/are

If $$\displaystyle\left ( 1+x \right )^{n}=\sum_{r=0}^{n}C_{r}x^{r}$$ and $$\sum { \sum _{ 0\le i<j\le n }{ { C }_{ i }\times { C }_{ j } }  } $$ represent the products of the $$C_{i}$$'s taken two at a time, then its value equals

Sum of the coefficients of the terms of degree $$m$$ in the expansion of
$${ (1+x) }^{ n }{ (1+y) }^{ n }{ (1+z) }^{ n }$$ is

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

In the expansion of $$ \displaystyle \left ( \sqrt{2} + \sqrt[5]{3}\right )^{120} $$ the number of irrational terms is