Binomial Theorem

In the expansion of $$(1 + x)^n$$ by the increasing powers of $$x$$, the third term is four times as great as the fifth term, and the ratio of the fourth term to the sixth is $$\dfrac {40}{3}$$. Find $$n$$ and $$x$$.

If log 1001= 3.000434, find the number of digits in $${1001^{101}}$$

Simplify the binomial $$\displaystyle\, \left ( \frac{x + 1}{x^{2/3} - x^{1/3} + 1} - \frac{x -1}{x - x^{1/2}} \right )^{10}$$ and find the term of its expansion which does not contain $$x$$.
Prove that the coefficient of $${ x }^{ p }$$ in the expansion of $${ \left( { x }^{ 2 }+\dfrac { 1 }{ x }  \right)  }^{ 2n }$$ is $$\dfrac { \left( 2n \right) ! }{ \left( \dfrac { 4n-p }{ 3 }  \right) !\left( \dfrac { 2n+p }{ 3 }  \right) ! }$$